Lp space - Wikipedia

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id="toc-The_p-norm_in_finite_dimensions-sublist" class="vector-toc-list"> <li id="toc-Relations_between_p-norms" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Relations_between_p-norms"> <div class="vector-toc-text"> 1.1.1 Relations between p-norms </div> </a> <ul id="toc-Relations_between_p-norms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-When_0_<_p_<_1" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#When_0_<_p_<_1"> <div class="vector-toc-text"> 1.1.2 When 0 < p < 1 </div> </a> <ul id="toc-When_0_<_p_<_1-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-When_p_=_0" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#When_p_=_0"> <div class="vector-toc-text"> 1.1.3 When p = 0 </div> </a> <ul id="toc-When_p_=_0-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-ℓp_spaces_and_sequence_spaces" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#ℓp_spaces_and_sequence_spaces"> <div class="vector-toc-text"> 2 ℓp spaces and sequence spaces </div> </a> <button aria-controls="toc-ℓp_spaces_and_sequence_spaces-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle ℓp spaces and sequence spaces subsection </button> <ul id="toc-ℓp_spaces_and_sequence_spaces-sublist" class="vector-toc-list"> <li id="toc-General_ℓp-space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_ℓp-space"> <div class="vector-toc-text"> 2.1 General ℓp-space </div> </a> <ul id="toc-General_ℓp-space-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Lp_spaces_and_Lebesgue_integrals" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Lp_spaces_and_Lebesgue_integrals"> <div class="vector-toc-text"> 3 Lp spaces and Lebesgue integrals </div> </a> <button aria-controls="toc-Lp_spaces_and_Lebesgue_integrals-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Lp spaces and Lebesgue integrals subsection </button> <ul id="toc-Lp_spaces_and_Lebesgue_integrals-sublist" class="vector-toc-list"> <li id="toc-Special_cases" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Special_cases"> <div class="vector-toc-text"> 3.1 Special cases </div> </a> <ul id="toc-Special_cases-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-When_(0_<_p_<_1)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#When_(0_<_p_<_1)"> <div class="vector-toc-text"> 3.2 When (0 < p < 1) </div> </a> <ul id="toc-When_(0_<_p_<_1)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> 4 Properties </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Properties subsection </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Hölder's_inequality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hölder's_inequality"> <div class="vector-toc-text"> 4.1 Hölder's inequality </div> </a> <ul id="toc-Hölder's_inequality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalized_Minkowski_inequality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalized_Minkowski_inequality"> <div class="vector-toc-text"> 4.2 Generalized Minkowski inequality </div> </a> <ul id="toc-Generalized_Minkowski_inequality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Atomic_decomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Atomic_decomposition"> <div class="vector-toc-text"> 4.3 Atomic decomposition </div> </a> <ul id="toc-Atomic_decomposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dual_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dual_spaces"> <div class="vector-toc-text"> 4.4 Dual spaces </div> </a> <ul id="toc-Dual_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Embeddings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Embeddings"> <div class="vector-toc-text"> 4.5 Embeddings </div> </a> <ul id="toc-Embeddings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dense_subspaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dense_subspaces"> <div class="vector-toc-text"> 4.6 Dense subspaces </div> </a> <ul id="toc-Dense_subspaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Closed_subspaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Closed_subspaces"> <div class="vector-toc-text"> 4.7 Closed subspaces </div> </a> <ul id="toc-Closed_subspaces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> 5 Applications </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Applications subsection </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Statistics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Statistics"> <div class="vector-toc-text"> 5.1 Statistics </div> </a> <ul id="toc-Statistics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hausdorff–Young_inequality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hausdorff–Young_inequality"> <div class="vector-toc-text"> 5.2 Hausdorff–Young inequality </div> </a> <ul id="toc-Hausdorff–Young_inequality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hilbert_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hilbert_spaces"> <div class="vector-toc-text"> 5.3 Hilbert spaces </div> </a> <ul id="toc-Hilbert_spaces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations_and_extensions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalizations_and_extensions"> <div class="vector-toc-text"> 6 Generalizations and extensions </div> </a> <button aria-controls="toc-Generalizations_and_extensions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Generalizations and extensions subsection </button> <ul id="toc-Generalizations_and_extensions-sublist" class="vector-toc-list"> <li id="toc-Weak_Lp" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Weak_Lp"> <div class="vector-toc-text"> 6.1 Weak Lp </div> </a> <ul id="toc-Weak_Lp-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Weighted_Lp_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Weighted_Lp_spaces"> <div class="vector-toc-text"> 6.2 Weighted Lp spaces </div> </a> <ul id="toc-Weighted_Lp_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lp_spaces_on_manifolds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lp_spaces_on_manifolds"> <div class="vector-toc-text"> 6.3 Lp spaces on manifolds </div> </a> <ul id="toc-Lp_spaces_on_manifolds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vector-valued_Lp_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vector-valued_Lp_spaces"> <div class="vector-toc-text"> 6.4 Vector-valued Lp spaces </div> </a> <ul id="toc-Vector-valued_Lp_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-L0_space_of_measurable_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#L0_space_of_measurable_functions"> <div class="vector-toc-text"> 6.5 L0 space of measurable functions </div> </a> <ul id="toc-L0_space_of_measurable_functions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 7 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 8 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 9 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 10 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the 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Available in 25 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-25" aria-hidden="true" > 25 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%D8%A1_%D8%A5%D9%84_%D8%A8%D9%8A" title="فضاء إل بي – Arabic" lang="ar" hreflang="ar" data-title="فضاء إل بي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Espai_Lp" title="Espai Lp – Catalan" lang="ca" hreflang="ca" data-title="Espai Lp" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Lp_prostor" title="Lp prostor – Czech" lang="cs" hreflang="cs" data-title="Lp prostor" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Lp_(matematik)" title="Lp (matematik) – Danish" lang="da" hreflang="da" data-title="Lp (matematik)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Lp-Raum" title="Lp-Raum – German" lang="de" hreflang="de" data-title="Lp-Raum" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Espacios_Lp" title="Espacios Lp – Spanish" lang="es" hreflang="es" data-title="Espacios Lp" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%DB%8C_%D9%84%D8%A8%DA%AF" title="فضای لبگ – Persian" lang="fa" hreflang="fa" data-title="فضای لبگ" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Espace_Lp" title="Espace Lp – French" lang="fr" hreflang="fr" data-title="Espace Lp" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A5%B4%EB%B2%A0%EA%B7%B8_%EA%B3%B5%EA%B0%84" title="르베그 공간 – Korean" lang="ko" hreflang="ko" data-title="르베그 공간" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Spazio_Lp" title="Spazio Lp – Italian" lang="it" hreflang="it" data-title="Spazio Lp" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A8%D7%97%D7%91_Lp" title="מרחב Lp – Hebrew" lang="he" hreflang="he" data-title="מרחב Lp" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Lebego_erdv%C4%97" title="Lebego erdvė – Lithuanian" lang="lt" hreflang="lt" data-title="Lebego erdvė" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Lp-ruimte" title="Lp-ruimte – Dutch" lang="nl" hreflang="nl" data-title="Lp-ruimte" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/Lp%E7%A9%BA%E9%96%93" title="Lp空間 – Japanese" lang="ja" hreflang="ja" data-title="Lp空間" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Lp-rom" title="Lp-rom – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Lp-rom" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Spassi_Lp" title="Spassi Lp – Piedmontese" lang="pms" hreflang="pms" data-title="Spassi Lp" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przestrze%C5%84_Lp" title="Przestrzeń Lp – Polish" lang="pl" hreflang="pl" data-title="Przestrzeń Lp" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Espa%C3%A7o_Lp" title="Espaço Lp – Portuguese" lang="pt" hreflang="pt" data-title="Espaço Lp" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Spa%C8%9Biu_Lp" title="Spațiu Lp – Romanian" lang="ro" hreflang="ro" data-title="Spațiu Lp" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/Lp_(%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE)" title="Lp (пространство) – Russian" lang="ru" hreflang="ru" data-title="Lp (пространство)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Lp-avaruus" title="Lp-avaruus – Finnish" lang="fi" hreflang="fi" data-title="Lp-avaruus" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Lp-rum" title="Lp-rum – Swedish" lang="sv" hreflang="sv" data-title="Lp-rum" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Lp_uzay%C4%B1" title="Lp uzayı – Turkish" lang="tr" hreflang="tr" data-title="Lp uzayı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D1%96%D1%80_Lp" title="Простір Lp – Ukrainian" lang="uk" hreflang="uk" data-title="Простір Lp" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li 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title="Norm (mathematics)">p-norm</a> for finite-dimensional <a href="/wiki/Vector_space" title="Vector space">vector spaces</a>. They are sometimes called Lebesgue spaces, named after <a href="/wiki/Henri_Lebesgue" title="Henri Lebesgue">Henri Lebesgue</a> (<a href="#CITEREFDunfordSchwartz1958">Dunford & Schwartz 1958</a>, III.3), although according to the <a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki</a> group (<a href="#CITEREFBourbaki1987">Bourbaki 1987</a>) they were first introduced by <a href="/wiki/Frigyes_Riesz" title="Frigyes Riesz">Frigyes Riesz</a> (<a href="#CITEREFRiesz1910">Riesz 1910</a>). Lp spaces form an important class of <a href="/wiki/Banach_space" title="Banach space">Banach spaces</a> in <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>, and of <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector spaces</a>. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Preliminaries">Preliminaries</h2>[<a href="/w/index.php?title=Lp_space&action=edit&section=1" title="Edit section: Preliminaries">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="The_p-norm_in_finite_dimensions">The p-norm in finite dimensions</h3>[<a href="/w/index.php?title=Lp_space&action=edit&section=2" title="Edit section: The p-norm in finite dimensions">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Vector-p-Norms_qtl1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Vector-p-Norms_qtl1.svg/220px-Vector-p-Norms_qtl1.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Vector-p-Norms_qtl1.svg/330px-Vector-p-Norms_qtl1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Vector-p-Norms_qtl1.svg/440px-Vector-p-Norms_qtl1.svg.png 2x" data-file-width="410" data-file-height="410" /></a><figcaption>Illustrations of <a href="/wiki/Unit_circle" title="Unit circle">unit circles</a> (see also <a href="/wiki/Superellipse" title="Superellipse">superellipse</a>) in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"> based on different <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-norms (every vector from the origin to the unit circle has a length of one, the length being calculated with length-formula of the corresponding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">).</figcaption></figure> The Euclidean length of a vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=(x_{1},x_{2},\dots ,x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=(x_{1},x_{2},\dots ,x_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d750ee2fffa2be890b9626b672037f0104cfe7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.766ex; height:2.843ex;" alt="{\displaystyle x=(x_{1},x_{2},\dots ,x_{n})}"> in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional <a href="/wiki/Real_number" title="Real number">real</a> <a href="/wiki/Vector_space" title="Vector space">vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"> is given by the <a href="/wiki/Euclidean_norm" class="mw-redirect" title="Euclidean norm">Euclidean norm</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|_{2}=\left({x_{1}}^{2}+{x_{2}}^{2}+\dotsb +{x_{n}}^{2}\right)^{1/2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|_{2}=\left({x_{1}}^{2}+{x_{2}}^{2}+\dotsb +{x_{n}}^{2}\right)^{1/2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/189d9c830d0abe2bc271bab8781b5b8daf83721d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.005ex; height:3.843ex;" alt="{\displaystyle \|x\|_{2}=\left({x_{1}}^{2}+{x_{2}}^{2}+\dotsb +{x_{n}}^{2}\right)^{1/2}.}"> The Euclidean distance between two points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> is the length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x-y\|_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x-y\|_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f4c73443d5463a3c783ebf7f62b4a072417dc13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.705ex; height:2.843ex;" alt="{\displaystyle \|x-y\|_{2}}"> of the straight line between the two points. In many situations, the Euclidean distance is appropriate for capturing the actual distances in a given space. In contrast, consider taxi drivers in a grid street plan who should measure distance not in terms of the length of the straight line to their destination, but in terms of the <a href="/wiki/Taxicab_geometry" title="Taxicab geometry">rectilinear distance</a>, which takes into account that streets are either orthogonal or parallel to each other. The class of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-norms generalizes these two examples and has an abundance of applications in many parts of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <a href="/wiki/Physics" title="Physics">physics</a>, and <a href="/wiki/Computer_science" title="Computer science">computer science</a>. For a <a href="/wiki/Real_number" title="Real number">real number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\geq 1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>≥</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\geq 1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/807af4aba28eab9b14fead4001f0bf1f2b93cbba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.167ex; height:2.509ex;" alt="{\displaystyle p\geq 1,}"> the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-norm or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}">-norm of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> is defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dotsb +|x_{n}|^{p}\right)^{1/p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dotsb +|x_{n}|^{p}\right)^{1/p}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea54f2beb5a1e2579a7ee1f5a81b87ff6c7132bd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.59ex; height:3.676ex;" alt="{\displaystyle \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dotsb +|x_{n}|^{p}\right)^{1/p}.}"> The absolute value bars can be dropped when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> is a rational number with an even numerator in its reduced form, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> is drawn from the set of real numbers, or one of its subsets. The Euclidean norm from above falls into this class and is the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}">-norm, and the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}">-norm is the norm that corresponds to the <a href="/wiki/Taxicab_geometry" title="Taxicab geometry">rectilinear distance</a>. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ab400cc4dfd865180cd84c72dc894ca457671f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.458ex; height:2.343ex;" alt="{\displaystyle L^{\infty }}">-norm or <a href="/wiki/Chebyshev_distance" title="Chebyshev distance">maximum norm</a> (or uniform norm) is the limit of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}">-norms for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\to \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97e17a610b63143ad03345436192293e3c698cb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.197ex; height:2.176ex;" alt="{\displaystyle p\to \infty }">, given by: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|_{\infty }=\max \left\{|x_{1}|,|x_{2}|,\dotsc ,|x_{n}|\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|_{\infty }=\max \left\{|x_{1}|,|x_{2}|,\dotsc ,|x_{n}|\right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e0e94b08377178ce45af7b0ce5da28a01d1461b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.075ex; height:2.843ex;" alt="{\displaystyle \|x\|_{\infty }=\max \left\{|x_{1}|,|x_{2}|,\dotsc ,|x_{n}|\right\}}"> For all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\geq 1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>≥</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\geq 1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/807af4aba28eab9b14fead4001f0bf1f2b93cbba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.167ex; height:2.509ex;" alt="{\displaystyle p\geq 1,}"> the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-norms and maximum norm satisfy the properties of a "length function" (or <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a>), that is: <ul><li>only the zero vector has zero length,</li> <li>the length of the vector is positive homogeneous with respect to multiplication by a scalar (<a href="/wiki/Euler%27s_homogeneous_function_theorem" class="mw-redirect" title="Euler's homogeneous function theorem">positive homogeneity</a>), and</li> <li>the length of the sum of two vectors is no larger than the sum of lengths of the vectors (<a href="/wiki/Triangle_inequality" title="Triangle inequality">triangle inequality</a>).</li></ul> Abstractly speaking, this means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"> together with the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-norm is a <a href="/wiki/Normed_vector_space" title="Normed vector space">normed vector space</a>. Moreover, it turns out that this space is <a href="/wiki/Complete_metric_space" title="Complete metric space">complete</a>, thus making it a <a href="/wiki/Banach_space" title="Banach space">Banach space</a>. <div class="mw-heading mw-heading4"><h4 id="Relations_between_p-norms">Relations between p-norms</h4>[<a href="/w/index.php?title=Lp_space&action=edit&section=3" title="Edit section: Relations between p-norms">edit</a>]</div> The grid distance or rectilinear distance (sometimes called the "<a href="/wiki/Manhattan_distance" class="mw-redirect" title="Manhattan distance">Manhattan distance</a>") between two points is never shorter than the length of the line segment between them (the Euclidean or "as the crow flies" distance). Formally, this means that the Euclidean norm of any vector is bounded by its 1-norm: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|_{2}\leq \|x\|_{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|_{2}\leq \|x\|_{1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/174592dfddd3765715db770e84a9186277b42c9f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.163ex; height:2.843ex;" alt="{\displaystyle \|x\|_{2}\leq \|x\|_{1}.}"> This fact generalizes to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-norms in that the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-norm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03811b9e3d1471b0ee1d217cc750faf77eb42a4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.714ex; height:3.009ex;" alt="{\displaystyle \|x\|_{p}}"> of any given vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> does not grow with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">: <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent" style="padding-left: 1.5em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|_{p+a}\leq \|x\|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>+</mo> <mi>a</mi> </mrow> </msub> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|_{p+a}\leq \|x\|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d152ca75ec1199acca43eb8131d46002ef57c5a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.674ex; height:3.009ex;" alt="{\displaystyle \|x\|_{p+a}\leq \|x\|_{p}}"> for any vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> and real numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf93c1353080a21b276e79058d82c19c40310653" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p\geq 1}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\geq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≥</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\geq 0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c12fce6c863aa37e821db93bedc430d956c29895" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.138ex; height:2.343ex;" alt="{\displaystyle a\geq 0.}"> (In fact this remains true for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<p<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<p<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea074f5b36db6eff17f1aa84d73e30e3de12c4d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.691ex; height:2.509ex;" alt="{\displaystyle 0<p<1}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\geq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c33f2f7ac8b81e1296d581427bdb2d30a50a2b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.491ex; height:2.343ex;" alt="{\displaystyle a\geq 0}"> .)</div> For the opposite direction, the following relation between the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}">-norm and the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}">-norm is known: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|_{1}\leq {\sqrt {n}}\|x\|_{2}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|_{1}\leq {\sqrt {n}}\|x\|_{2}~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7e234e857aa13ce6dc59810cb692e707cfd07a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.074ex; height:3.009ex;" alt="{\displaystyle \|x\|_{1}\leq {\sqrt {n}}\|x\|_{2}~.}"> This inequality depends on the dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> of the underlying vector space and follows directly from the <a href="/wiki/Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz inequality</a>. In general, for vectors in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<r<p:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>r</mi> <mo><</mo> <mi>p</mi> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<r<p:}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfe3784ae9f51e9f485375e8e28d8e9e13795f8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.869ex; height:2.509ex;" alt="{\displaystyle 0<r<p:}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|_{p}\leq \|x\|_{r}\leq n^{{\frac {1}{r}}-{\frac {1}{p}}}\|x\|_{p}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>≤</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mrow> </mrow> </msup> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|_{p}\leq \|x\|_{r}\leq n^{{\frac {1}{r}}-{\frac {1}{p}}}\|x\|_{p}~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a381d2acb86864bd12189818ed6591ad4a5a754f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.397ex; height:4.176ex;" alt="{\displaystyle \|x\|_{p}\leq \|x\|_{r}\leq n^{{\frac {1}{r}}-{\frac {1}{p}}}\|x\|_{p}~.}"> This is a consequence of <a href="/wiki/H%C3%B6lder%27s_inequality" title="Hölder's inequality">Hölder's inequality</a>. <div class="mw-heading mw-heading4"><h4 id="When_0_<_p_<_1">When 0 < p < 1</h4>[<a href="/w/index.php?title=Lp_space&action=edit&section=4" title="Edit section: When 0 edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Astroid.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Astroid.svg/220px-Astroid.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Astroid.svg/330px-Astroid.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Astroid.svg/440px-Astroid.svg.png 2x" data-file-width="340" data-file-height="340" /></a><figcaption><a href="/wiki/Astroid" title="Astroid">Astroid</a>, unit circle in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p={\tfrac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\tfrac {2}{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ab69c822cd5f6119794478fa6065a22b6e521cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; margin-left: -0.089ex; width:6.015ex; height:3.676ex;" alt="{\displaystyle p={\tfrac {2}{3}}}"> metric</figcaption></figure> In <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/957e317312fd02f34c1d1c8c80bd8484c29fde6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.302ex; height:2.509ex;" alt="{\displaystyle n>1,}"> the formula <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\cdots +|x_{n}|^{p}\right)^{1/p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\cdots +|x_{n}|^{p}\right)^{1/p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/559ec2af54a144b631dd25f5d8497e43efe5d8ba" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.944ex; height:3.676ex;" alt="{\displaystyle \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\cdots +|x_{n}|^{p}\right)^{1/p}}"> defines an absolutely <a href="/wiki/Homogeneous_function" title="Homogeneous function">homogeneous function</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<p<1;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>1</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<p<1;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/113dfc6a10d7a2adddd23efc6aa863a65c61bf4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.338ex; height:2.509ex;" alt="{\displaystyle 0<p<1;}"> however, the resulting function does not define a norm, because it is not <a href="/wiki/Subadditivity" title="Subadditivity">subadditive</a>. On the other hand, the formula <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x_{1}|^{p}+|x_{2}|^{p}+\dotsb +|x_{n}|^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x_{1}|^{p}+|x_{2}|^{p}+\dotsb +|x_{n}|^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51d06e5a277d8e3b3fdf9caa8db834576e13b5fe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.619ex; height:3.009ex;" alt="{\displaystyle |x_{1}|^{p}+|x_{2}|^{p}+\dotsb +|x_{n}|^{p}}"> defines a subadditive function at the cost of losing absolute homogeneity. It does define an <a href="/wiki/F-space" title="F-space">F-norm</a>, though, which is homogeneous of degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88532f4eab1d4cef71ef96c0f8c98cac36fd9257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.906ex; height:2.009ex;" alt="{\displaystyle p.}"> Hence, the function <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{p}(x,y)=\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{p}(x,y)=\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83ccad8b3b8f31e398a28df3ce819288e49be189" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.699ex; height:6.843ex;" alt="{\displaystyle d_{p}(x,y)=\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}}"> defines a <a href="/wiki/Metric_space" title="Metric space">metric</a>. The <a href="/wiki/Metric_space" title="Metric space">metric space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {R} ^{n},d_{p})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {R} ^{n},d_{p})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ea92b0a6e86741eef65fbcd652b5f56e80ae215" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.008ex; height:3.009ex;" alt="{\displaystyle (\mathbb {R} ^{n},d_{p})}"> is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{n}^{p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{n}^{p}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/820c189e2726ea42b64a822af89200e8638575ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.835ex; height:2.843ex;" alt="{\displaystyle \ell _{n}^{p}.}"> Although the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-unit ball <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee78f22d82d3b185b98644405bc0ca5662976739" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.843ex;" alt="{\displaystyle B_{n}^{p}}"> around the origin in this metric is "concave", the topology defined on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"> by the metric <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18a1069ed15b2551691c1f85039842d7e7642f05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.823ex; height:2.843ex;" alt="{\displaystyle B_{p}}"> is the usual vector space topology of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7035fcb9fe3ebecc6bc9f372f82d0352202c8bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{n},}"> hence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{n}^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{n}^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cb1b2cdb3afa507e7c086ab38ababe717f652eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.188ex; height:2.843ex;" alt="{\displaystyle \ell _{n}^{p}}"> is a <a href="/wiki/Locally_convex" class="mw-redirect" title="Locally convex">locally convex</a> topological vector space. Beyond this qualitative statement, a quantitative way to measure the lack of convexity of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{n}^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{n}^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cb1b2cdb3afa507e7c086ab38ababe717f652eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.188ex; height:2.843ex;" alt="{\displaystyle \ell _{n}^{p}}"> is to denote by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{p}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{p}(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/254117dfa6373dafc3c94469a3a781b065297aa2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.925ex; height:3.009ex;" alt="{\displaystyle C_{p}(n)}"> the smallest constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> such that the scalar multiple <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\,B_{n}^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mspace width="thinmathspace" /> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C\,B_{n}^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38c7bf7de523ae9776c6f4d367085d15e45da515" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.136ex; height:2.843ex;" alt="{\displaystyle C\,B_{n}^{p}}"> of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-unit ball contains the convex hull of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}^{p},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}^{p},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/983bde1d41bf31f1eeb70bd7369763f294e5455b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.629ex; height:2.843ex;" alt="{\displaystyle B_{n}^{p},}"> which is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}^{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}^{1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205e58d8e549438b48c9a6aaa6946973e4f80648" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.629ex; height:2.843ex;" alt="{\displaystyle B_{n}^{1}.}"> The fact that for fixed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f40fda4df30e9f9f3354c808cc122e5dadbc2d99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p<1}"> we have <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{p}(n)=n^{{\tfrac {1}{p}}-1}\to \infty ,\quad {\text{as }}n\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mstyle> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>as </mtext> </mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{p}(n)=n^{{\tfrac {1}{p}}-1}\to \infty ,\quad {\text{as }}n\to \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d713cb45a61a55bbb03c7d4ddb8a1bef281ff71" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.7ex; height:4.843ex;" alt="{\displaystyle C_{p}(n)=n^{{\tfrac {1}{p}}-1}\to \infty ,\quad {\text{as }}n\to \infty }"> shows that the infinite-dimensional sequence space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c352341ab7260ca1a9004da44c897d1395203c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.029ex; height:2.343ex;" alt="{\displaystyle \ell ^{p}}"> defined below, is no longer locally convex.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <div class="mw-heading mw-heading4"><h4 id="When_p_=_0">When p = 0</h4>[<a href="/w/index.php?title=Lp_space&action=edit&section=5" title="Edit section: When p = 0">edit</a>]</div> There is one <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d18f7cb79dd41b63d6aca9ec6b957c225a0aea81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.024ex; height:2.509ex;" alt="{\displaystyle \ell _{0}}"> norm and another function called the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d18f7cb79dd41b63d6aca9ec6b957c225a0aea81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.024ex; height:2.509ex;" alt="{\displaystyle \ell _{0}}"> "norm" (with quotation marks). The mathematical definition of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d18f7cb79dd41b63d6aca9ec6b957c225a0aea81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.024ex; height:2.509ex;" alt="{\displaystyle \ell _{0}}"> norm was established by <a href="/wiki/Stefan_Banach" title="Stefan Banach">Banach</a>'s <a href="/w/index.php?title=Theory_of_Linear_Operations&action=edit&redlink=1" class="new" title="Theory of Linear Operations (page does not exist)">Theory of Linear Operations</a>. The <a href="/wiki/F-space" title="F-space">space</a> of sequences has a complete metric topology provided by the <a href="/wiki/F-space" title="F-space">F-norm</a> on the <a href="/wiki/Metric_space#Product_metric_spaces" title="Metric space">product metric</a>:[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{n})\mapsto \|x\|:=d(0,x)=\sum _{n}2^{-n}{\frac {|x_{n}|}{1+|x_{n}|}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>:=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{n})\mapsto \|x\|:=d(0,x)=\sum _{n}2^{-n}{\frac {|x_{n}|}{1+|x_{n}|}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e017db36e66f5b8ab679c34d9382f19b3d5330f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.75ex; height:6.843ex;" alt="{\displaystyle (x_{n})\mapsto \|x\|:=d(0,x)=\sum _{n}2^{-n}{\frac {|x_{n}|}{1+|x_{n}|}}.}"> The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d18f7cb79dd41b63d6aca9ec6b957c225a0aea81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.024ex; height:2.509ex;" alt="{\displaystyle \ell _{0}}">-normed space is studied in functional analysis, probability theory, and harmonic analysis. Another function was called the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d18f7cb79dd41b63d6aca9ec6b957c225a0aea81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.024ex; height:2.509ex;" alt="{\displaystyle \ell _{0}}"> "norm" by <a href="/wiki/David_Donoho" title="David Donoho">David Donoho</a>—whose quotation marks warn that this function is not a proper norm—is the number of non-zero entries of the vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d07e9f568a88785ae48006ac3c4b951020f1699a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.977ex; height:1.676ex;" alt="{\displaystyle x.}">[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] Many authors <a href="/wiki/Abuse_of_terminology" class="mw-redirect" title="Abuse of terminology">abuse terminology</a> by omitting the quotation marks. Defining <a href="/wiki/Zero_to_the_power_of_zero" title="Zero to the power of zero"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{0}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{0}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79bb8d19ac3fb8851d89034bee612fa0ed1842c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.124ex; height:3.009ex;" alt="{\displaystyle 0^{0}=0,}"></a> the zero "norm" of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> is equal to <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x_{1}|^{0}+|x_{2}|^{0}+\cdots +|x_{n}|^{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x_{1}|^{0}+|x_{2}|^{0}+\cdots +|x_{n}|^{0}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2e1f30c0eed3cbbcf1c20c9d07b7f43f1a2e0de" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.251ex; height:3.343ex;" alt="{\displaystyle |x_{1}|^{0}+|x_{2}|^{0}+\cdots +|x_{n}|^{0}.}"> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Lp_space_animation.gif" class="mw-file-description"><img alt="An animated gif of p-norms 0.1 through 2 with a step of 0.05." src="//upload.wikimedia.org/wikipedia/commons/1/1f/Lp_space_animation.gif" decoding="async" width="64" height="64" class="mw-file-element" data-file-width="64" data-file-height="64" /></a><figcaption>An animated gif of p-norms 0.1 through 2 with a step of 0.05.</figcaption></figure> This is not a <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a> because it is not <a href="/wiki/Homogeneous_function" title="Homogeneous function">homogeneous</a>. For example, scaling the vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> by a positive constant does not change the "norm". Despite these defects as a mathematical norm, the non-zero counting "norm" has uses in <a href="/wiki/Scientific_computing" class="mw-redirect" title="Scientific computing">scientific computing</a>, <a href="/wiki/Information_theory" title="Information theory">information theory</a>, and <a href="/wiki/Statistics" title="Statistics">statistics</a>–notably in <a href="/wiki/Compressed_sensing" title="Compressed sensing">compressed sensing</a> in <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a> and computational <a href="/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a>. Despite not being a norm, the associated metric, known as <a href="/wiki/Hamming_distance" title="Hamming distance">Hamming distance</a>, is a valid distance, since homogeneity is not required for distances. <div class="mw-heading mw-heading2"><h2 id="ℓp_spaces_and_sequence_spaces">ℓp spaces and sequence spaces</h2>[<a href="/w/index.php?title=Lp_space&action=edit&section=6" title="Edit section: ℓp spaces and sequence spaces">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Sequence_space" title="Sequence space">Sequence space</a></div> The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-norm can be extended to vectors that have an infinite number of components (<a href="/wiki/Sequence" title="Sequence">sequences</a>), which yields the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{p}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de985eb969ec4307cee491238aa9aa19fc54c7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.676ex; height:2.343ex;" alt="{\displaystyle \ell ^{p}.}"> This contains as special cases: <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{1},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03ededdd80a2eaaf2c4b55f0effe594d389a7646" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.671ex; height:3.009ex;" alt="{\displaystyle \ell ^{1},}"> the space of sequences whose series are <a href="/wiki/Absolute_convergence" title="Absolute convergence">absolutely convergent</a>,</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/086013af803aa5075639eec27825b24ea35b4fb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.671ex; height:3.009ex;" alt="{\displaystyle \ell ^{2},}"> the space of square-summable sequences, which is a <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a>, and</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6522774415b5d36e38cf2bf77f346263c876400" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.492ex; height:2.676ex;" alt="{\displaystyle \ell ^{\infty },}"> the space of <a href="/wiki/Bounded_sequence" class="mw-redirect" title="Bounded sequence">bounded sequences</a>.</li></ul> The space of sequences has a natural vector space structure by applying scalar addition and multiplication. Explicitly, the vector sum and the scalar action for infinite <a href="/wiki/Sequence" title="Sequence">sequences</a> of real (or <a href="/wiki/Complex_number" title="Complex number">complex</a>) numbers are given by: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&(x_{1},x_{2},\ldots ,x_{n},x_{n+1},\ldots )+(y_{1},y_{2},\ldots ,y_{n},y_{n+1},\ldots )\\={}&(x_{1}+y_{1},x_{2}+y_{2},\ldots ,x_{n}+y_{n},x_{n+1}+y_{n+1},\ldots ),\\[6pt]&\lambda \cdot \left(x_{1},x_{2},\ldots ,x_{n},x_{n+1},\ldots \right)\\={}&(\lambda x_{1},\lambda x_{2},\ldots ,\lambda x_{n},\lambda x_{n+1},\ldots ).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt 0.9em 0.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi></mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mi></mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi>λ</mi> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mi></mi> <mo stretchy="false">(</mo> <mi>λ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>λ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>λ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mi>λ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&(x_{1},x_{2},\ldots ,x_{n},x_{n+1},\ldots )+(y_{1},y_{2},\ldots ,y_{n},y_{n+1},\ldots )\\={}&(x_{1}+y_{1},x_{2}+y_{2},\ldots ,x_{n}+y_{n},x_{n+1}+y_{n+1},\ldots ),\\[6pt]&\lambda \cdot \left(x_{1},x_{2},\ldots ,x_{n},x_{n+1},\ldots \right)\\={}&(\lambda x_{1},\lambda x_{2},\ldots ,\lambda x_{n},\lambda x_{n+1},\ldots ).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/511c06a4932ed1b5013fa317b49b92424b97413f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:54.838ex; height:13.843ex;" alt="{\displaystyle {\begin{aligned}&(x_{1},x_{2},\ldots ,x_{n},x_{n+1},\ldots )+(y_{1},y_{2},\ldots ,y_{n},y_{n+1},\ldots )\\={}&(x_{1}+y_{1},x_{2}+y_{2},\ldots ,x_{n}+y_{n},x_{n+1}+y_{n+1},\ldots ),\\[6pt]&\lambda \cdot \left(x_{1},x_{2},\ldots ,x_{n},x_{n+1},\ldots \right)\\={}&(\lambda x_{1},\lambda x_{2},\ldots ,\lambda x_{n},\lambda x_{n+1},\ldots ).\end{aligned}}}"> Define the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-norm: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\cdots +|x_{n}|^{p}+|x_{n+1}|^{p}+\cdots \right)^{1/p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\cdots +|x_{n}|^{p}+|x_{n+1}|^{p}+\cdots \right)^{1/p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7641a6d55df29b5b9dc359147b52671aeaf18cae" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:53.349ex; height:3.676ex;" alt="{\displaystyle \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\cdots +|x_{n}|^{p}+|x_{n+1}|^{p}+\cdots \right)^{1/p}}"> Here, a complication arises, namely that the <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a> on the right is not always convergent, so for example, the sequence made up of only ones, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,1,1,\ldots ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,1,1,\ldots ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9cfb517eca8f74abc0bfd22bbe60cdd259ad5d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.769ex; height:2.843ex;" alt="{\displaystyle (1,1,1,\ldots ),}"> will have an infinite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-norm for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p<\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq p<\infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e169aecd0a30936992e97b8fe62aa114854954d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.499ex; height:2.509ex;" alt="{\displaystyle 1\leq p<\infty .}"> The space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c352341ab7260ca1a9004da44c897d1395203c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.029ex; height:2.343ex;" alt="{\displaystyle \ell ^{p}}"> is then defined as the set of all infinite sequences of real (or complex) numbers such that the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-norm is finite. One can check that as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> increases, the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c352341ab7260ca1a9004da44c897d1395203c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.029ex; height:2.343ex;" alt="{\displaystyle \ell ^{p}}"> grows larger. For example, the sequence <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(1,{\frac {1}{2}},\ldots ,{\frac {1}{n}},{\frac {1}{n+1}},\ldots \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(1,{\frac {1}{2}},\ldots ,{\frac {1}{n}},{\frac {1}{n+1}},\ldots \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7c169240a3bda874fce8cbd673a41e47c58bf7d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.05ex; height:6.176ex;" alt="{\displaystyle \left(1,{\frac {1}{2}},\ldots ,{\frac {1}{n}},{\frac {1}{n+1}},\ldots \right)}"> is not in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{1},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03ededdd80a2eaaf2c4b55f0effe594d389a7646" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.671ex; height:3.009ex;" alt="{\displaystyle \ell ^{1},}"> but it is in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c352341ab7260ca1a9004da44c897d1395203c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.029ex; height:2.343ex;" alt="{\displaystyle \ell ^{p}}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p>1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>></mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p>1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b18ad57c656bab68c7bd45e2bf4596d09de7c0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.167ex; height:2.509ex;" alt="{\displaystyle p>1,}"> as the series <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1^{p}+{\frac {1}{2^{p}}}+\cdots +{\frac {1}{n^{p}}}+{\frac {1}{(n+1)^{p}}}+\cdots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1^{p}+{\frac {1}{2^{p}}}+\cdots +{\frac {1}{n^{p}}}+{\frac {1}{(n+1)^{p}}}+\cdots ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cadfd9d45d5307debecabfda60f7133c8abcc5a8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.354ex; height:6.009ex;" alt="{\displaystyle 1^{p}+{\frac {1}{2^{p}}}+\cdots +{\frac {1}{n^{p}}}+{\frac {1}{(n+1)^{p}}}+\cdots ,}"> diverges for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c29a2f2fb3f642618036ed7a79712202e7ada924" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=1}"> (the <a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">harmonic series</a>), but is convergent for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p>1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>></mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p>1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/150fa9ba91afdd519352dd06ca4ca262989c5ae8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.167ex; height:2.509ex;" alt="{\displaystyle p>1.}"> One also defines the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }">-norm using the <a href="/wiki/Supremum" class="mw-redirect" title="Supremum">supremum</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|_{\infty }=\sup(|x_{1}|,|x_{2}|,\dotsc ,|x_{n}|,|x_{n+1}|,\ldots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">sup</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|_{\infty }=\sup(|x_{1}|,|x_{2}|,\dotsc ,|x_{n}|,|x_{n+1}|,\ldots )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f20dc5120426d9ef8e99eecc21d74fcf274be163" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.082ex; height:2.843ex;" alt="{\displaystyle \|x\|_{\infty }=\sup(|x_{1}|,|x_{2}|,\dotsc ,|x_{n}|,|x_{n+1}|,\ldots )}"> and the corresponding space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8348195cf09473662c6f59e6717722a6fc01d0f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.845ex; height:2.343ex;" alt="{\displaystyle \ell ^{\infty }}"> of all bounded sequences. It turns out that<a href="#cite_note-1">[1]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|_{\infty }=\lim _{p\to \infty }\|x\|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|_{\infty }=\lim _{p\to \infty }\|x\|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e61078e70d6481a02d673a1f8619d01e6dc0805d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.843ex; height:4.176ex;" alt="{\displaystyle \|x\|_{\infty }=\lim _{p\to \infty }\|x\|_{p}}"> if the right-hand side is finite, or the left-hand side is infinite. Thus, we will consider <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c352341ab7260ca1a9004da44c897d1395203c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.029ex; height:2.343ex;" alt="{\displaystyle \ell ^{p}}"> spaces for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p\leq \infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq p\leq \infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58b3ded74395e6ce45c63e6e290599f32b506799" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.499ex; height:2.509ex;" alt="{\displaystyle 1\leq p\leq \infty .}"> The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-norm thus defined on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c352341ab7260ca1a9004da44c897d1395203c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.029ex; height:2.343ex;" alt="{\displaystyle \ell ^{p}}"> is indeed a norm, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c352341ab7260ca1a9004da44c897d1395203c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.029ex; height:2.343ex;" alt="{\displaystyle \ell ^{p}}"> together with this norm is a <a href="/wiki/Banach_space" title="Banach space">Banach space</a>. <div class="mw-heading mw-heading3"><h3 id="General_ℓp-space">General ℓp-space</h3>[<a href="/w/index.php?title=Lp_space&action=edit&section=7" title="Edit section: General ℓp-space">edit</a>]</div> In complete analogy to the preceding definition one can define the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{p}(I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{p}(I)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c617935ca4e181c8fc9114d9549febd88d86c720" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.01ex; height:2.843ex;" alt="{\displaystyle \ell ^{p}(I)}"> over a general <a href="/wiki/Index_set" title="Index set">index set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> (and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq p<\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d42661a6d23dcde0fb7ae1b611e233958d13da9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.853ex; height:2.509ex;" alt="{\displaystyle 1\leq p<\infty }">) as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{p}(I)=\left\{(x_{i})_{i\in I}\in \mathbb {K} ^{I}:\sum _{i\in I}|x_{i}|^{p}<+\infty \right\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msup> <mo>:</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo><</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{p}(I)=\left\{(x_{i})_{i\in I}\in \mathbb {K} ^{I}:\sum _{i\in I}|x_{i}|^{p}<+\infty \right\},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8eb681b1a314ff232446379879778b366dfc777d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.653ex; height:7.509ex;" alt="{\displaystyle \ell ^{p}(I)=\left\{(x_{i})_{i\in I}\in \mathbb {K} ^{I}:\sum _{i\in I}|x_{i}|^{p}<+\infty \right\},}"> where convergence on the right means that only countably many summands are nonzero (see also <a href="/wiki/Unconditional_convergence" title="Unconditional convergence">Unconditional convergence</a>). With the norm <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|_{p}=\left(\sum _{i\in I}|x_{i}|^{p}\right)^{1/p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|_{p}=\left(\sum _{i\in I}|x_{i}|^{p}\right)^{1/p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fe5cb50e5b2302e4fcba28fbb92e8fc96d14d0d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.421ex; height:8.009ex;" alt="{\displaystyle \|x\|_{p}=\left(\sum _{i\in I}|x_{i}|^{p}\right)^{1/p}}"> the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{p}(I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{p}(I)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c617935ca4e181c8fc9114d9549febd88d86c720" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.01ex; height:2.843ex;" alt="{\displaystyle \ell ^{p}(I)}"> becomes a Banach space. In the case where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> is finite with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> elements, this construction yields <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"> with the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-norm defined above. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> is countably infinite, this is exactly the sequence space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c352341ab7260ca1a9004da44c897d1395203c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.029ex; height:2.343ex;" alt="{\displaystyle \ell ^{p}}"> defined above. For uncountable sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> this is a non-<a href="/wiki/Separable_space" title="Separable space">separable</a> Banach space which can be seen as the <a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">locally convex</a> <a href="/wiki/Direct_limit" title="Direct limit">direct limit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c352341ab7260ca1a9004da44c897d1395203c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.029ex; height:2.343ex;" alt="{\displaystyle \ell ^{p}}">-sequence spaces.<a href="#cite_note-2">[2]</a> For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=2,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e497adeac79e22e49edc8c064b15e3b7d899c9a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.167ex; height:2.509ex;" alt="{\displaystyle p=2,}"> the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\,\cdot \,\|_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mspace width="thinmathspace" /> <mo>⋅</mo> <mspace width="thinmathspace" /> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\,\cdot \,\|_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f70a78163331d7880f3831701fcb709341a2798e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.832ex; height:2.843ex;" alt="{\displaystyle \|\,\cdot \,\|_{2}}">-norm is even induced by a canonical <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \,\cdot ,\,\cdot \rangle ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mspace width="thinmathspace" /> <mo>⋅</mo> <mo>,</mo> <mspace width="thinmathspace" /> <mo>⋅</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \,\cdot ,\,\cdot \rangle ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db488043739d54e3bad7fd10165f41eb45491aa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.558ex; height:2.843ex;" alt="{\displaystyle \langle \,\cdot ,\,\cdot \rangle ,}"> called the <style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style>Euclidean inner product, which means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\mathbf {x} \|_{2}={\sqrt {\langle \mathbf {x} ,\mathbf {x} \rangle }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {x} \|_{2}={\sqrt {\langle \mathbf {x} ,\mathbf {x} \rangle }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b2ef6cbe5b0679ea387928890f95e1250c09097" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.878ex; height:4.843ex;" alt="{\displaystyle \|\mathbf {x} \|_{2}={\sqrt {\langle \mathbf {x} ,\mathbf {x} \rangle }}}"> holds for all vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ab831ab2afdb20ebc1acde8fbb8352032f68d01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.058ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} .}"> This inner product can expressed in terms of the norm by using the <a href="/wiki/Polarization_identity" title="Polarization identity">polarization identity</a>. On <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/086013af803aa5075639eec27825b24ea35b4fb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.671ex; height:3.009ex;" alt="{\displaystyle \ell ^{2},}"> it can be defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \left(x_{i}\right)_{i},\left(y_{n}\right)_{i}\rangle _{\ell ^{2}}~=~\sum _{i}x_{i}{\overline {y_{i}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \left(x_{i}\right)_{i},\left(y_{n}\right)_{i}\rangle _{\ell ^{2}}~=~\sum _{i}x_{i}{\overline {y_{i}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5010beace529958be0b320ad6db1b86c94fc82f1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.517ex; height:5.509ex;" alt="{\displaystyle \langle \left(x_{i}\right)_{i},\left(y_{n}\right)_{i}\rangle _{\ell ^{2}}~=~\sum _{i}x_{i}{\overline {y_{i}}}.}"> Now consider the case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=\infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e84b68700f76da4a8befa9b0b17662b9daca968" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.328ex; height:2.009ex;" alt="{\displaystyle p=\infty .}"> Define<a href="#cite_note-3">[note 1]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }(I)=\{x\in \mathbb {K} ^{I}:\sup \operatorname {range} |x|<+\infty \},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">K</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msup> <mo>:</mo> <mo movablelimits="true" form="prefix">sup</mo> <mi>range</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }(I)=\{x\in \mathbb {K} ^{I}:\sup \operatorname {range} |x|<+\infty \},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8864ebf0d6dfa497ced3a836c334edb71b54864d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.563ex; height:3.176ex;" alt="{\displaystyle \ell ^{\infty }(I)=\{x\in \mathbb {K} ^{I}:\sup \operatorname {range} |x|<+\infty \},}"> where for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"><a href="#cite_note-4">[3]</a><a href="#cite_note-5">[note 2]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|_{\infty }\equiv \inf\{C\in \mathbb {R} _{\geq 0}:|x_{i}|\leq C{\text{ for all }}i\in I\}={\begin{cases}\sup \operatorname {range} |x|&{\text{if }}X\neq \varnothing ,\\0&{\text{if }}X=\varnothing .\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>≡</mo> <mo movablelimits="true" form="prefix">inf</mo> <mo fence="false" stretchy="false">{</mo> <mi>C</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>≥</mo> <mn>0</mn> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>i</mi> <mo>∈</mo> <mi>I</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo movablelimits="true" form="prefix">sup</mo> <mi>range</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>X</mi> <mo>≠</mo> <mi class="MJX-variant">∅</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>X</mi> <mo>=</mo> <mi class="MJX-variant">∅</mi> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|_{\infty }\equiv \inf\{C\in \mathbb {R} _{\geq 0}:|x_{i}|\leq C{\text{ for all }}i\in I\}={\begin{cases}\sup \operatorname {range} |x|&{\text{if }}X\neq \varnothing ,\\0&{\text{if }}X=\varnothing .\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfcad03c13fbf154b50979a95b9043079a02ea8f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:74.095ex; height:6.176ex;" alt="{\displaystyle \|x\|_{\infty }\equiv \inf\{C\in \mathbb {R} _{\geq 0}:|x_{i}|\leq C{\text{ for all }}i\in I\}={\begin{cases}\sup \operatorname {range} |x|&{\text{if }}X\neq \varnothing ,\\0&{\text{if }}X=\varnothing .\end{cases}}}"> The index set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> can be turned into a <a href="/wiki/Measure_space" title="Measure space">measure space</a> by giving it the <a href="/wiki/%CE%A3-algebra#Simple_set-based_examples" title="Σ-algebra">discrete σ-algebra</a> and the <a href="/wiki/Counting_measure" title="Counting measure">counting measure</a>. Then the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{p}(I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{p}(I)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c617935ca4e181c8fc9114d9549febd88d86c720" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.01ex; height:2.843ex;" alt="{\displaystyle \ell ^{p}(I)}"> is just a special case of the more general <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}">-space (defined below). <div class="mw-heading mw-heading2"><h2 id="Lp_spaces_and_Lebesgue_integrals">Lp spaces and Lebesgue integrals</h2>[<a href="/w/index.php?title=Lp_space&action=edit&section=8" title="Edit section: Lp spaces and Lebesgue integrals">edit</a>]</div> An <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"> space may be defined as a space of measurable functions for which the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-th power of the <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> is <a href="/wiki/Lebesgue_integrable" class="mw-redirect" title="Lebesgue integrable">Lebesgue integrable</a>, where functions which agree almost everywhere are identified. More generally, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S,\Sigma ,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S,\Sigma ,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cf1dfd7f18980638e424a78b4f5fe40df597c8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.456ex; height:2.843ex;" alt="{\displaystyle (S,\Sigma ,\mu )}"> be a <a href="/wiki/Measure_space" title="Measure space">measure space</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p\leq \infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq p\leq \infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58b3ded74395e6ce45c63e6e290599f32b506799" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.499ex; height:2.509ex;" alt="{\displaystyle 1\leq p\leq \infty .}"><a href="#cite_note-6">[note 3]</a> When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\neq \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>≠</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\neq \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6e8a56e74706550119d79a997e495328828cc65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.681ex; height:2.676ex;" alt="{\displaystyle p\neq \infty }">, consider the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44f87cc85cf5190b48fe6c61c049b0199f68a4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.794ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}"> of all <a href="/wiki/Measurable_function" title="Measurable function">measurable functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> whose <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> raised to the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-th power has a finite integral, or in symbols:<a href="#cite_note-FOOTNOTERudin198765-7">[4]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{p}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left(\int _{S}|f|^{p}\;\mathrm {d} \mu \right)^{1/p}<\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="2"> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mstyle> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>μ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msup> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{p}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left(\int _{S}|f|^{p}\;\mathrm {d} \mu \right)^{1/p}<\infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65b100e6e239a6a95e2de6176e843c9bfc6c5951" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.768ex; height:6.676ex;" alt="{\displaystyle \|f\|_{p}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left(\int _{S}|f|^{p}\;\mathrm {d} \mu \right)^{1/p}<\infty .}"> To define the set for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=\infty ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=\infty ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcf616a43b37b2de8dae9af2cf7ec5828055c51f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.328ex; height:2.009ex;" alt="{\displaystyle p=\infty ,}"> recall that two functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> defined on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> are said to be equal <a href="/wiki/Almost_everywhere" title="Almost everywhere">almost everywhere</a>, written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/795e79b6da5372a37ba3a36db68e43806232aac3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.493ex; height:2.509ex;" alt="{\displaystyle f=g}"> a.e., if the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{s\in S:f(s)\neq g(s)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>s</mi> <mo>∈</mo> <mi>S</mi> <mo>:</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{s\in S:f(s)\neq g(s)\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69f6af6ee82fb411e4f9282d7350a7ddb1d58f59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.985ex; height:2.843ex;" alt="{\displaystyle \{s\in S:f(s)\neq g(s)\}}"> is measurable and has measure zero. Similarly, a measurable function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> (and its <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a>) is bounded (or dominated) almost everywhere by a real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64528f031cdbe1f52bdaf4ba7a8401108c0d2dc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.413ex; height:2.509ex;" alt="{\displaystyle C,}"> written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f|\leq C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f|\leq C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/021582ab59421dd23372b5e1fe6823272c0a0527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.437ex; height:2.843ex;" alt="{\displaystyle |f|\leq C}"> a.e., if the (necessarily) measurable set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{s\in S:|f(s)|>C\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>s</mi> <mo>∈</mo> <mi>S</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mi>C</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{s\in S:|f(s)|>C\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf6daacc8da9d93ca8629a86c0702120561628ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.029ex; height:2.843ex;" alt="{\displaystyle \{s\in S:|f(s)|>C\}}"> has measure zero. The space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}^{\infty }(S,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}^{\infty }(S,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd1b810e06de816204b210dc3fd5c816244de2ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.223ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}^{\infty }(S,\mu )}"> is the set of all measurable functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> that are bounded almost everywhere (by some real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}">) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{\infty }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7e68102540dd0b3c823d7710ba31aa8b72cf7cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.479ex; height:2.843ex;" alt="{\displaystyle \|f\|_{\infty }}"> is defined as the <a href="/wiki/Infimum" class="mw-redirect" title="Infimum">infimum</a> of these bounds: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{\infty }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\inf\{C\in \mathbb {R} _{\geq 0}:|f(s)|\leq C{\text{ for almost every }}s\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="2"> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mstyle> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mo movablelimits="true" form="prefix">inf</mo> <mo fence="false" stretchy="false">{</mo> <mi>C</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>≥</mo> <mn>0</mn> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> for almost every </mtext> </mrow> <mi>s</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{\infty }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\inf\{C\in \mathbb {R} _{\geq 0}:|f(s)|\leq C{\text{ for almost every }}s\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71f2766418b041d5beed63b5aeae2122b5c076fc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.709ex; height:3.509ex;" alt="{\displaystyle \|f\|_{\infty }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\inf\{C\in \mathbb {R} _{\geq 0}:|f(s)|\leq C{\text{ for almost every }}s\}.}"> When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (S)\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (S)\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cfcf941bff745f2789de102419a2da3fd8b4d2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.971ex; height:2.843ex;" alt="{\displaystyle \mu (S)\neq 0}"> then this is the same as the <a href="/wiki/Essential_supremum" class="mw-redirect" title="Essential supremum">essential supremum</a> of the absolute value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">:<a href="#cite_note-8">[note 4]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{\infty }~=~{\begin{cases}\operatorname {esssup} |f|&{\text{if }}\mu (S)>0,\\0&{\text{if }}\mu (S)=0.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>esssup</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{\infty }~=~{\begin{cases}\operatorname {esssup} |f|&{\text{if }}\mu (S)>0,\\0&{\text{if }}\mu (S)=0.\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e826bd60fcf1628700e4a018d8902c892f962cad" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.439ex; height:6.176ex;" alt="{\displaystyle \|f\|_{\infty }~=~{\begin{cases}\operatorname {esssup} |f|&{\text{if }}\mu (S)>0,\\0&{\text{if }}\mu (S)=0.\end{cases}}}"> For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is a measurable function that is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> almost everywhere<a href="#cite_note-Non0Value0Example-9">[note 5]</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{p}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{p}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40bdbb097cc1abe5e06afde70f65f0de59fa1188" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.924ex; height:3.009ex;" alt="{\displaystyle \|f\|_{p}=0}"> for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> and thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in {\mathcal {L}}^{p}(S,\,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in {\mathcal {L}}^{p}(S,\,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/319b2dbe9576dbadebb8771aada5c960641cb6af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.913ex; height:2.843ex;" alt="{\displaystyle f\in {\mathcal {L}}^{p}(S,\,\mu )}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88532f4eab1d4cef71ef96c0f8c98cac36fd9257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.906ex; height:2.009ex;" alt="{\displaystyle p.}"> For every positive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/393fcf18074cb42eafb26b76c515a1e93e17512c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.906ex; height:2.009ex;" alt="{\displaystyle p,}"> the value under <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\,\cdot \,\|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mspace width="thinmathspace" /> <mo>⋅</mo> <mspace width="thinmathspace" /> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\,\cdot \,\|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b38aecbe3d80d669e37b8232614a63a09df2e9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.837ex; height:3.009ex;" alt="{\displaystyle \|\,\cdot \,\|_{p}}"> of a measurable function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> and its absolute value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f|:S\to [0,\infty ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>:</mo> <mi>S</mi> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f|:S\to [0,\infty ]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d9853e0278b1d78ff17d84ebc8c16bd23783cf5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.437ex; height:2.843ex;" alt="{\displaystyle |f|:S\to [0,\infty ]}"> are always the same (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{p}=\||f|\|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{p}=\||f|\|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/034c317477d85bdb5bca042c0b09feb12466911b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.717ex; height:3.009ex;" alt="{\displaystyle \|f\|_{p}=\||f|\|_{p}}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">) and so a measurable function belongs to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44f87cc85cf5190b48fe6c61c049b0199f68a4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.794ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}"> if and only if its absolute value does. Because of this, many formulas involving <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-norms are stated only for non-negative real-valued functions. Consider for example the identity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{p}^{r}=\|f^{r}\|_{p/r},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{p}^{r}=\|f^{r}\|_{p/r},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b7b1ab61fb360137c68ea1e744afe79aa2bf42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.65ex; height:3.176ex;" alt="{\displaystyle \|f\|_{p}^{r}=\|f^{r}\|_{p/r},}"> which holds whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\geq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/696698f05d2d7595339e163685632f64732cc55c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.509ex;" alt="{\displaystyle f\geq 0}"> is measurable, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23cbbcd53bd13620bc53490e3eec42790850b452" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r>0}"> is real, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<p\leq \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo>≤</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<p\leq \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/915ecfea83a5b7e3432407d4eebea5a38f9948ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.853ex; height:2.509ex;" alt="{\displaystyle 0<p\leq \infty }"> (here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty /r\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="2"> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mstyle> </mrow> </mover> </mrow> </mrow> <mspace width="thickmathspace" /> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty /r\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d25fdc56654023bc52ff503789932c2d0be5803c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.957ex; height:3.509ex;" alt="{\displaystyle \infty /r\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\infty }"> when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42846cbf8fc7db1ff19e51d0d43ca84c249b062b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.681ex; height:2.009ex;" alt="{\displaystyle p=\infty }">). The non-negativity requirement <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\geq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/696698f05d2d7595339e163685632f64732cc55c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.509ex;" alt="{\displaystyle f\geq 0}"> can be removed by substituting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/940e58aa437f429f47fd743a819e41fedaa1ff9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.572ex; height:2.843ex;" alt="{\displaystyle |f|}"> in for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e9687ea22c0f310582e97ee5f6c6a5fca28203d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.925ex; height:2.509ex;" alt="{\displaystyle f,}"> which gives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\,|f|\,\|_{p}^{r}=\|\,|f|^{r}\,\|_{p/r}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\,|f|\,\|_{p}^{r}=\|\,|f|^{r}\,\|_{p/r}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a486b3deb91ffc32e0b9566be9c57156531fa19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.744ex; height:3.343ex;" alt="{\displaystyle \|\,|f|\,\|_{p}^{r}=\|\,|f|^{r}\,\|_{p/r}.}"> Note in particular that when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69f2db560d595689d7a7276cddbceacffa24b064" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.406ex; height:2.009ex;" alt="{\displaystyle p=r}"> is finite then the formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{p}^{p}=\||f|^{p}\|_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{p}^{p}=\||f|^{p}\|_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66b3fb1837445eda7cc57aca28a9bf4729a11be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.772ex; height:3.176ex;" alt="{\displaystyle \|f\|_{p}^{p}=\||f|^{p}\|_{1}}"> relates the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-norm to the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}">-norm. Seminormed space of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-th power integrable functions Each set of functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44f87cc85cf5190b48fe6c61c049b0199f68a4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.794ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}"> forms a <a href="/wiki/Vector_space" title="Vector space">vector space</a> when addition and scalar multiplication are defined pointwise.<a href="#cite_note-10">[note 6]</a> That the sum of two <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-th power integrable functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> is again <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-th power integrable follows from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \|f+g\|_{p}^{p}\leq 2^{p-1}\left(\|f\|_{p}^{p}+\|g\|_{p}^{p}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mo>+</mo> <mi>g</mi> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <mo>≤</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>g</mi> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \|f+g\|_{p}^{p}\leq 2^{p-1}\left(\|f\|_{p}^{p}+\|g\|_{p}^{p}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d792cb5a9f41f80e79890a416ac089eea589f20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.273ex; height:3.343ex;" alt="{\textstyle \|f+g\|_{p}^{p}\leq 2^{p-1}\left(\|f\|_{p}^{p}+\|g\|_{p}^{p}\right),}"><a href="#cite_note-UpperBoundForNormOfSum-11">[proof 1]</a> although it is also a consequence of <a href="/wiki/Minkowski_inequality" title="Minkowski inequality">Minkowski's inequality</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f+g\|_{p}\leq \|f\|_{p}+\|g\|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mo>+</mo> <mi>g</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>g</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f+g\|_{p}\leq \|f\|_{p}+\|g\|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1d3de762058656808721fc899c4b223914c6c3f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.721ex; height:3.009ex;" alt="{\displaystyle \|f+g\|_{p}\leq \|f\|_{p}+\|g\|_{p}}"> which establishes that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44e951f6e15fdb1bdd7d223bbbe628955251a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.063ex; height:3.009ex;" alt="{\displaystyle \|\cdot \|_{p}}"> satisfies the <a href="/wiki/Triangle_inequality" title="Triangle inequality">triangle inequality</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p\leq \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq p\leq \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7d63bb8c8def80f4fb709fbb2aae6a11c9cd41d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.853ex; height:2.509ex;" alt="{\displaystyle 1\leq p\leq \infty }"> (the triangle inequality does not hold for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<p<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<p<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea074f5b36db6eff17f1aa84d73e30e3de12c4d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.691ex; height:2.509ex;" alt="{\displaystyle 0<p<1}">). That <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44f87cc85cf5190b48fe6c61c049b0199f68a4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.794ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}"> is closed under scalar multiplication is due to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44e951f6e15fdb1bdd7d223bbbe628955251a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.063ex; height:3.009ex;" alt="{\displaystyle \|\cdot \|_{p}}"> being <a href="/wiki/Absolute_homogeneity" class="mw-redirect" title="Absolute homogeneity">absolutely homogeneous</a>, which means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|sf\|_{p}=|s|\|f\|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>s</mi> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|sf\|_{p}=|s|\|f\|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/615e895c611035f7fa5ec931fa4d04b59beea0b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.898ex; height:3.009ex;" alt="{\displaystyle \|sf\|_{p}=|s|\|f\|_{p}}"> for every scalar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"> and every function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecb3ed2e17fa8f336dcc0fd4b3eddbfb02a50ef3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.925ex; height:2.509ex;" alt="{\displaystyle f.}"> <a href="/wiki/Absolute_homogeneity" class="mw-redirect" title="Absolute homogeneity">Absolute homogeneity</a>, the <a href="/wiki/Triangle_inequality" title="Triangle inequality">triangle inequality</a>, and non-negativity are the defining properties of a <a href="/wiki/Seminorm" title="Seminorm">seminorm</a>. Thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44e951f6e15fdb1bdd7d223bbbe628955251a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.063ex; height:3.009ex;" alt="{\displaystyle \|\cdot \|_{p}}"> is a seminorm and the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44f87cc85cf5190b48fe6c61c049b0199f68a4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.794ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-th power integrable functions together with the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44e951f6e15fdb1bdd7d223bbbe628955251a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.063ex; height:3.009ex;" alt="{\displaystyle \|\cdot \|_{p}}"> defines a <a href="/wiki/Seminormed_vector_space" class="mw-redirect" title="Seminormed vector space">seminormed vector space</a>. In general, the <a href="/wiki/Seminorm" title="Seminorm">seminorm</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44e951f6e15fdb1bdd7d223bbbe628955251a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.063ex; height:3.009ex;" alt="{\displaystyle \|\cdot \|_{p}}"> is not a <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a> because there might exist measurable functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> that satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{p}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{p}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40bdbb097cc1abe5e06afde70f65f0de59fa1188" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.924ex; height:3.009ex;" alt="{\displaystyle \|f\|_{p}=0}"> but are not identically equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"><a href="#cite_note-Non0Value0Example-9">[note 5]</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44e951f6e15fdb1bdd7d223bbbe628955251a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.063ex; height:3.009ex;" alt="{\displaystyle \|\cdot \|_{p}}"> is a norm if and only if no such <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> exists). Zero sets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-seminorms If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is measurable and equals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> a.e. then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{p}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{p}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40bdbb097cc1abe5e06afde70f65f0de59fa1188" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.924ex; height:3.009ex;" alt="{\displaystyle \|f\|_{p}=0}"> for all positive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\leq \infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>≤</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\leq \infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e4600e17e423a31f73242d0fad0cf8d8e9b24d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.328ex; height:2.343ex;" alt="{\displaystyle p\leq \infty .}"> On the other hand, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is a measurable function for which there exists some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<p\leq \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo>≤</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<p\leq \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/915ecfea83a5b7e3432407d4eebea5a38f9948ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.853ex; height:2.509ex;" alt="{\displaystyle 0<p\leq \infty }"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{p}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{p}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40bdbb097cc1abe5e06afde70f65f0de59fa1188" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.924ex; height:3.009ex;" alt="{\displaystyle \|f\|_{p}=0}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee0fdf0f50fcba5afe3e856fcc7dc6acfa61014" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.509ex;" alt="{\displaystyle f=0}"> almost everywhere. When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> is finite then this follows from the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c29a2f2fb3f642618036ed7a79712202e7ada924" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=1}"> case and the formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{p}^{p}=\||f|^{p}\|_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{p}^{p}=\||f|^{p}\|_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66b3fb1837445eda7cc57aca28a9bf4729a11be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.772ex; height:3.176ex;" alt="{\displaystyle \|f\|_{p}^{p}=\||f|^{p}\|_{1}}"> mentioned above. Thus if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\leq \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>≤</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\leq \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ced467164fc991031784c1592f0ec1abbe4fb22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.681ex; height:2.343ex;" alt="{\displaystyle p\leq \infty }"> is positive and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is any measurable function, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{p}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{p}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40bdbb097cc1abe5e06afde70f65f0de59fa1188" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.924ex; height:3.009ex;" alt="{\displaystyle \|f\|_{p}=0}"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee0fdf0f50fcba5afe3e856fcc7dc6acfa61014" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.509ex;" alt="{\displaystyle f=0}"> <a href="/wiki/Almost_everywhere" title="Almost everywhere">almost everywhere</a>. Since the right hand side (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee0fdf0f50fcba5afe3e856fcc7dc6acfa61014" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.509ex;" alt="{\displaystyle f=0}"> a.e.) does not mention <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/393fcf18074cb42eafb26b76c515a1e93e17512c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.906ex; height:2.009ex;" alt="{\displaystyle p,}"> it follows that all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44e951f6e15fdb1bdd7d223bbbe628955251a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.063ex; height:3.009ex;" alt="{\displaystyle \|\cdot \|_{p}}"> have the same <a href="/wiki/Zero_set" class="mw-redirect" title="Zero set">zero set</a> (it does not depend on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">). So denote this common set by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\{f:f=0\ \mu {\text{-almost everywhere}}\}=\{f\in {\mathcal {L}}^{p}(S,\,\mu ):\|f\|_{p}=0\}\qquad \forall \ p.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="2"> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mstyle> </mrow> </mover> </mrow> </mrow> <mspace width="thickmathspace" /> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>:</mo> <mi>f</mi> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>-almost everywhere</mtext> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mspace width="2em" /> <mi mathvariant="normal">∀</mi> <mtext> </mtext> <mi>p</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\{f:f=0\ \mu {\text{-almost everywhere}}\}=\{f\in {\mathcal {L}}^{p}(S,\,\mu ):\|f\|_{p}=0\}\qquad \forall \ p.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe3a48c9f03e819f3994509cce85827bc5407d7c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.062ex; width:75.492ex; height:3.676ex;" alt="{\displaystyle {\mathcal {N}}\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\{f:f=0\ \mu {\text{-almost everywhere}}\}=\{f\in {\mathcal {L}}^{p}(S,\,\mu ):\|f\|_{p}=0\}\qquad \forall \ p.}"> This set is a vector subspace of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44f87cc85cf5190b48fe6c61c049b0199f68a4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.794ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}"> for every positive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\leq \infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>≤</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\leq \infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e4600e17e423a31f73242d0fad0cf8d8e9b24d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.328ex; height:2.343ex;" alt="{\displaystyle p\leq \infty .}"> Quotient vector space Like every <a href="/wiki/Seminorm" title="Seminorm">seminorm</a>, the seminorm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44e951f6e15fdb1bdd7d223bbbe628955251a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.063ex; height:3.009ex;" alt="{\displaystyle \|\cdot \|_{p}}"> induces a <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a> (defined shortly) on the canonical <a href="/wiki/Quotient_space_(linear_algebra)" title="Quotient space (linear algebra)">quotient vector space</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44f87cc85cf5190b48fe6c61c049b0199f68a4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.794ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}"> by its vector subspace <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {N}}=\{f\in {\mathcal {L}}^{p}(S,\,\mu ):\|f\|_{p}=0\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathcal {N}}=\{f\in {\mathcal {L}}^{p}(S,\,\mu ):\|f\|_{p}=0\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dab79d872b61b1c101a75b4528e1fc6f34d621bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.062ex; width:32.181ex; height:3.176ex;" alt="{\textstyle {\mathcal {N}}=\{f\in {\mathcal {L}}^{p}(S,\,\mu ):\|f\|_{p}=0\}.}"> This normed quotient space is called Lebesgue space and it is the subject of this article. We begin by defining the quotient vector space. Given any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in {\mathcal {L}}^{p}(S,\,\mu ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in {\mathcal {L}}^{p}(S,\,\mu ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b15dc79fff1355330652129657b8f436c48cf96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.56ex; height:2.843ex;" alt="{\displaystyle f\in {\mathcal {L}}^{p}(S,\,\mu ),}"> the <a href="/wiki/Coset" title="Coset">coset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f+{\mathcal {N}}\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\{f+h:h\in {\mathcal {N}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="2"> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mstyle> </mrow> </mover> </mrow> </mrow> <mspace width="thickmathspace" /> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>+</mo> <mi>h</mi> <mo>:</mo> <mi>h</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f+{\mathcal {N}}\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\{f+h:h\in {\mathcal {N}}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a9177394169cb98ce138d7e8c9c65f2b3a74881" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.667ex; height:3.509ex;" alt="{\displaystyle f+{\mathcal {N}}\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\{f+h:h\in {\mathcal {N}}\}}"> consists of all measurable functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> that are equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> <a href="/wiki/Almost_everywhere" title="Almost everywhere">almost everywhere</a>. The set of all cosets, typically denoted by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}^{p}(S,\mu )/{\mathcal {N}}~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~~\{f+{\mathcal {N}}:f\in {\mathcal {L}}^{p}(S,\mu )\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="2"> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mstyle> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mtext> </mtext> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo>:</mo> <mi>f</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}^{p}(S,\mu )/{\mathcal {N}}~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~~\{f+{\mathcal {N}}:f\in {\mathcal {L}}^{p}(S,\mu )\},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c87d2f24bc856414baf613b600d346b7f7cfc37" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.804ex; height:3.509ex;" alt="{\displaystyle {\mathcal {L}}^{p}(S,\mu )/{\mathcal {N}}~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~~\{f+{\mathcal {N}}:f\in {\mathcal {L}}^{p}(S,\mu )\},}"> forms a vector space with origin <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0+{\mathcal {N}}={\mathcal {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0+{\mathcal {N}}={\mathcal {N}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a915f8e466d9b8c1f81db37e57cd0cdf9a45892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.651ex; height:2.676ex;" alt="{\displaystyle 0+{\mathcal {N}}={\mathcal {N}}}"> when vector addition and scalar multiplication are defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f+{\mathcal {N}})+(g+{\mathcal {N}})\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;(f+g)+{\mathcal {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="2"> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mstyle> </mrow> </mover> </mrow> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>f</mi> <mo>+</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f+{\mathcal {N}})+(g+{\mathcal {N}})\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;(f+g)+{\mathcal {N}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b4dae495ddb97d0b3a5447c9bb3fc0331766b95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.342ex; height:3.509ex;" alt="{\displaystyle (f+{\mathcal {N}})+(g+{\mathcal {N}})\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;(f+g)+{\mathcal {N}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(f+{\mathcal {N}})\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;(sf)+{\mathcal {N}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="2"> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mstyle> </mrow> </mover> </mrow> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>s</mi> <mi>f</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(f+{\mathcal {N}})\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;(sf)+{\mathcal {N}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/995065d7973f1c7a479b536c064a0ddc1c8ac3a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.333ex; height:3.509ex;" alt="{\displaystyle s(f+{\mathcal {N}})\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;(sf)+{\mathcal {N}}.}"> This particular quotient vector space will be denoted by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(S,\,\mu )~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\mathcal {L}}^{p}(S,\mu )/{\mathcal {N}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>μ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="2"> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mstyle> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(S,\,\mu )~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\mathcal {L}}^{p}(S,\mu )/{\mathcal {N}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5313b123653dc629aea22e507e42199365b3a7a6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.234ex; height:3.509ex;" alt="{\displaystyle L^{p}(S,\,\mu )~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\mathcal {L}}^{p}(S,\mu )/{\mathcal {N}}.}"> Two cosets are equal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f+{\mathcal {N}}=g+{\mathcal {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo>=</mo> <mi>g</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f+{\mathcal {N}}=g+{\mathcal {N}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f826cc5825f41848f077ab1a1990ef97ad55f43a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.724ex; height:2.843ex;" alt="{\displaystyle f+{\mathcal {N}}=g+{\mathcal {N}}}"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in f+{\mathcal {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∈</mo> <mi>f</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\in f+{\mathcal {N}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c17d96eb5ebe99a34ae04b3fb7641b8b4132f518" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.351ex; height:2.843ex;" alt="{\displaystyle g\in f+{\mathcal {N}}}"> (or equivalently, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f-g\in {\mathcal {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>−</mo> <mi>g</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f-g\in {\mathcal {N}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dac3e725ee32850a3b7dd1fdbf58390f2b455547" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.351ex; height:2.843ex;" alt="{\displaystyle f-g\in {\mathcal {N}}}">), which happens if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/795e79b6da5372a37ba3a36db68e43806232aac3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.493ex; height:2.509ex;" alt="{\displaystyle f=g}"> almost everywhere; if this is the case then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> are identified in the quotient space. Hence, strictly speaking <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(S,\,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(S,\,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0806c3e125a89e0704774b4db9944b9c4f01bfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.773ex; height:2.843ex;" alt="{\displaystyle L^{p}(S,\,\mu )}"> consists of <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence classes</a> of functions.<a href="#cite_note-FOOTNOTESteinShakarchi20122-12">[5]</a> The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-norm on the quotient vector space Given any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in {\mathcal {L}}^{p}(S,\,\mu ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in {\mathcal {L}}^{p}(S,\,\mu ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b15dc79fff1355330652129657b8f436c48cf96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.56ex; height:2.843ex;" alt="{\displaystyle f\in {\mathcal {L}}^{p}(S,\,\mu ),}"> the value of the seminorm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44e951f6e15fdb1bdd7d223bbbe628955251a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.063ex; height:3.009ex;" alt="{\displaystyle \|\cdot \|_{p}}"> on the <a href="/wiki/Coset" title="Coset">coset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f+{\mathcal {N}}=\{f+h:h\in {\mathcal {N}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>+</mo> <mi>h</mi> <mo>:</mo> <mi>h</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f+{\mathcal {N}}=\{f+h:h\in {\mathcal {N}}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d7931b8a2cc0a4decc0481cce421fcdccdf0aa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.667ex; height:3.009ex;" alt="{\displaystyle f+{\mathcal {N}}=\{f+h:h\in {\mathcal {N}}\}}"> is constant and equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{p};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{p};}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3df90cdc2f1beb3a113e1033a9ea8aca5b7e41e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.31ex; height:3.009ex;" alt="{\displaystyle \|f\|_{p};}"> denote this unique value by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f+{\mathcal {N}}\|_{p},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f+{\mathcal {N}}\|_{p},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f87e579f3de03492c27d2b209e0d01c259f2d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.425ex; height:3.176ex;" alt="{\displaystyle \|f+{\mathcal {N}}\|_{p},}"> so that: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f+{\mathcal {N}}\|_{p}\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\|f\|_{p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="2"> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mstyle> </mrow> </mover> </mrow> </mrow> <mspace width="thickmathspace" /> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f+{\mathcal {N}}\|_{p}\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\|f\|_{p}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ceeff26968157ee85657e7aec88a664311d6626" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.186ex; height:3.676ex;" alt="{\displaystyle \|f+{\mathcal {N}}\|_{p}\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\|f\|_{p}.}"> This assignment <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f+{\mathcal {N}}\mapsto \|f+{\mathcal {N}}\|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">↦</mo> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f+{\mathcal {N}}\mapsto \|f+{\mathcal {N}}\|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62bf60c04f00718aa3b6ee52a9ad030f6b895949" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.786ex; height:3.176ex;" alt="{\displaystyle f+{\mathcal {N}}\mapsto \|f+{\mathcal {N}}\|_{p}}"> defines a map, which will also be denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{p},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{p},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/084d3b2c0095344e5c4ec2ebeb8ca233f854dc43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.71ex; height:3.009ex;" alt="{\displaystyle \|\cdot \|_{p},}"> on the <a href="/wiki/Quotient_space_(linear_algebra)" title="Quotient space (linear algebra)">quotient vector space</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(S,\mu )~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~~{\mathcal {L}}^{p}(S,\mu )/{\mathcal {N}}~=~\{f+{\mathcal {N}}:f\in {\mathcal {L}}^{p}(S,\mu )\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="2"> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mstyle> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo>:</mo> <mi>f</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(S,\mu )~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~~{\mathcal {L}}^{p}(S,\mu )/{\mathcal {N}}~=~\{f+{\mathcal {N}}:f\in {\mathcal {L}}^{p}(S,\mu )\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/216931b9899343332744b3d118b5c6ab4e3f31a3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.45ex; height:3.509ex;" alt="{\displaystyle L^{p}(S,\mu )~~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~~{\mathcal {L}}^{p}(S,\mu )/{\mathcal {N}}~=~\{f+{\mathcal {N}}:f\in {\mathcal {L}}^{p}(S,\mu )\}.}"> This map is a <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(S,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(S,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d5a4c54613c1c773a1caa81a731c340f546e320" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.386ex; height:2.843ex;" alt="{\displaystyle L^{p}(S,\mu )}"> called the <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-norm. The value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f+{\mathcal {N}}\|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f+{\mathcal {N}}\|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c880d10783c7d14aca7ead435d43eb408f0fe901" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.778ex; height:3.176ex;" alt="{\displaystyle \|f+{\mathcal {N}}\|_{p}}"> of a coset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f+{\mathcal {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f+{\mathcal {N}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4333fe83d24f776255ed8965d9be48a38d6834f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.394ex; height:2.843ex;" alt="{\displaystyle f+{\mathcal {N}}}"> is independent of the particular function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> that was chosen to represent the coset, meaning that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}\in L^{p}(S,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\in L^{p}(S,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f08477671184c8675d0fa2a693a0e0ec6ad8514d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.466ex; height:2.843ex;" alt="{\displaystyle {\mathcal {C}}\in L^{p}(S,\mu )}"> is any coset then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|{\mathcal {C}}\|_{p}=\|f\|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|{\mathcal {C}}\|_{p}=\|f\|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/769ef3974c23adcc7b28c4b64aa78186deface7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.384ex; height:3.009ex;" alt="{\displaystyle \|{\mathcal {C}}\|_{p}=\|f\|_{p}}"> for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in {\mathcal {C}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f229e0bcf172e8ff39fe758851a495d2e5674dd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.358ex; height:2.509ex;" alt="{\displaystyle f\in {\mathcal {C}}}"> (since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}=f+{\mathcal {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mo>=</mo> <mi>f</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}=f+{\mathcal {N}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c0a50b509b6b2acbafa44ba10787d5b090f52de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.732ex; height:2.843ex;" alt="{\displaystyle {\mathcal {C}}=f+{\mathcal {N}}}"> for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in {\mathcal {C}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f229e0bcf172e8ff39fe758851a495d2e5674dd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.358ex; height:2.509ex;" alt="{\displaystyle f\in {\mathcal {C}}}">). The Lebesgue <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"> space The <a href="/wiki/Normed_vector_space" title="Normed vector space">normed vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(L^{p}(S,\mu ),\|\cdot \|_{p}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(L^{p}(S,\mu ),\|\cdot \|_{p}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b8c15ea491316e6af4116966a5e10f336f3a4f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.293ex; height:3.009ex;" alt="{\displaystyle \left(L^{p}(S,\mu ),\|\cdot \|_{p}\right)}"> is called <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"> space or the Lebesgue space of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-th power integrable functions and it is a <a href="/wiki/Banach_space" title="Banach space">Banach space</a> for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p\leq \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq p\leq \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7d63bb8c8def80f4fb709fbb2aae6a11c9cd41d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.853ex; height:2.509ex;" alt="{\displaystyle 1\leq p\leq \infty }"> (meaning that it is a <a href="/wiki/Complete_metric_space" title="Complete metric space">complete metric space</a>, a result that is sometimes called the <a href="/wiki/Riesz%E2%80%93Fischer_theorem#Completeness_of_Lp,_0_<_p_≤_∞" title="Riesz–Fischer theorem">Riesz–Fischer theorem</a>). When the underlying measure space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> is understood then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(S,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(S,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d5a4c54613c1c773a1caa81a731c340f546e320" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.386ex; height:2.843ex;" alt="{\displaystyle L^{p}(S,\mu )}"> is often abbreviated <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\mu ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\mu ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e4be03c6efe4321ddefe14c15a17dfcd85252e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.5ex; height:2.843ex;" alt="{\displaystyle L^{p}(\mu ),}"> or even just <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b13f0f126183015da10056f10da88534132b5ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.289ex; height:2.343ex;" alt="{\displaystyle L^{p}.}"> Depending on the author, the subscript notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e40060f367b8045868f06b72a6600969b07acaed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.642ex; height:2.843ex;" alt="{\displaystyle L_{p}}"> might denote either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(S,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(S,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d5a4c54613c1c773a1caa81a731c340f546e320" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.386ex; height:2.843ex;" alt="{\displaystyle L^{p}(S,\mu )}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1/p}(S,\mu ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{1/p}(S,\mu ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/084d548afb53daa510d78064a052daa30f1ad320" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.677ex; height:3.343ex;" alt="{\displaystyle L^{1/p}(S,\mu ).}"> If the seminorm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44e951f6e15fdb1bdd7d223bbbe628955251a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.063ex; height:3.009ex;" alt="{\displaystyle \|\cdot \|_{p}}"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44f87cc85cf5190b48fe6c61c049b0199f68a4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.794ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}"> happens to be a norm (which happens if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}=\{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}=\{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4db2bc6a7228576a93d2b1141f0f86633528dde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:8.922ex; height:3.009ex;" alt="{\displaystyle {\mathcal {N}}=\{0\}}">) then the normed space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\mathcal {L}}^{p}(S,\,\mu ),\|\cdot \|_{p}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\mathcal {L}}^{p}(S,\,\mu ),\|\cdot \|_{p}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dbca9b0818031ff02be14900ad890c217a9c43c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.701ex; height:3.009ex;" alt="{\displaystyle \left({\mathcal {L}}^{p}(S,\,\mu ),\|\cdot \|_{p}\right)}"> will be <a href="/wiki/Linear_map" title="Linear map">linearly</a> <a href="/wiki/Isometrically_isomorphic" class="mw-redirect" title="Isometrically isomorphic">isometrically isomorphic</a> to the normed quotient space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(L^{p}(S,\mu ),\|\cdot \|_{p}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(L^{p}(S,\mu ),\|\cdot \|_{p}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b8c15ea491316e6af4116966a5e10f336f3a4f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.293ex; height:3.009ex;" alt="{\displaystyle \left(L^{p}(S,\mu ),\|\cdot \|_{p}\right)}"> via the canonical map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in {\mathcal {L}}^{p}(S,\,\mu )\mapsto \{g\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>μ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mo fence="false" stretchy="false">{</mo> <mi>g</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\in {\mathcal {L}}^{p}(S,\,\mu )\mapsto \{g\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2c4a1338e853d636981296fc8654e3dff9ae6d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.806ex; height:2.843ex;" alt="{\displaystyle g\in {\mathcal {L}}^{p}(S,\,\mu )\mapsto \{g\}}"> (since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g+{\mathcal {N}}=\{g\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>g</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g+{\mathcal {N}}=\{g\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dffdacb3032addaa9db48a47e9cda830b9f3efe8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.771ex; height:3.009ex;" alt="{\displaystyle g+{\mathcal {N}}=\{g\}}">); in other words, they will be, <a href="/wiki/Up_to" title="Up to">up to</a> a <a href="/wiki/Linear_isometry" class="mw-redirect" title="Linear isometry">linear isometry</a>, the same normed space and so they may both be called "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"> space". The above definitions generalize to <a href="/wiki/Bochner_space" title="Bochner space">Bochner spaces</a>. In general, this process cannot be reversed: there is no consistent way to define a "canonical" representative of each coset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7551c7bed2cd2ee83e10536d157c94a5f8f72fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b13f0f126183015da10056f10da88534132b5ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.289ex; height:2.343ex;" alt="{\displaystyle L^{p}.}"> For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1164381464cd05921aff337d11cfc56793bb688" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.105ex; height:2.676ex;" alt="{\displaystyle L^{\infty },}"> however, there is a <a href="/wiki/Lifting_theory" title="Lifting theory">theory of lifts</a> enabling such recovery. <div class="mw-heading mw-heading3"><h3 id="Special_cases">Special cases</h3>[<a href="/w/index.php?title=Lp_space&action=edit&section=9" title="Edit section: Special cases">edit</a>]</div> For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p\leq \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq p\leq \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7d63bb8c8def80f4fb709fbb2aae6a11c9cd41d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.853ex; height:2.509ex;" alt="{\displaystyle 1\leq p\leq \infty }"> the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c352341ab7260ca1a9004da44c897d1395203c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.029ex; height:2.343ex;" alt="{\displaystyle \ell ^{p}}"> spaces are a special case of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"> spaces; when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> are the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> is the <a href="/wiki/Counting_measure" title="Counting measure">counting measure</a>. More generally, if one considers any set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> with the counting measure, the resulting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"> space is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{p}(S).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{p}(S).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398d027fd2d6fc593c0248f789c46e98531d0802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.984ex; height:2.843ex;" alt="{\displaystyle \ell ^{p}(S).}"> For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{p}(\mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{p}(\mathbb {Z} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9383f909a044dbd7873bf93e6f192df8f9a1c901" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.388ex; height:2.843ex;" alt="{\displaystyle \ell ^{p}(\mathbb {Z} )}"> is the space of all sequences indexed by the integers, and when defining the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-norm on such a space, one sums over all the integers. The space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{p}(n),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{p}(n),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ccf30eb73f98330cfedcd36f52390efe816d17b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.88ex; height:2.843ex;" alt="{\displaystyle \ell ^{p}(n),}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> is the set with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> elements, is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"> with its <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-norm as defined above. Similar to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91f1f909abd70bd3d8fff0f7ae1ac23052387e18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{2}}"> spaces, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba162c66ca85776c83557af5088cc6f8584d1912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}"> is the only <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> among <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"> spaces. In the complex case, the inner product on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba162c66ca85776c83557af5088cc6f8584d1912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}"> is defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle f,g\rangle =\int _{S}f(x){\overline {g(x)}}\,\mathrm {d} \mu (x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle f,g\rangle =\int _{S}f(x){\overline {g(x)}}\,\mathrm {d} \mu (x).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bda4dd84a9c0fe49dea717be3700ef420be46b48" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.963ex; height:5.676ex;" alt="{\displaystyle \langle f,g\rangle =\int _{S}f(x){\overline {g(x)}}\,\mathrm {d} \mu (x).}"> Functions in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba162c66ca85776c83557af5088cc6f8584d1912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}"> are sometimes called <a href="/wiki/Square-integrable_function" title="Square-integrable function">square-integrable functions</a>, quadratically integrable functions or square-summable functions, but sometimes these terms are reserved for functions that are square-integrable in some other sense, such as in the sense of a <a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a> (<a href="#CITEREFTitchmarsh1976">Titchmarsh 1976</a>). As any Hilbert space, every space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba162c66ca85776c83557af5088cc6f8584d1912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}"> is linearly isometric to a suitable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2}(I),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{2}(I),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6affe102026888a94ae80fd4de61b53e6be20112" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.652ex; height:3.176ex;" alt="{\displaystyle \ell ^{2}(I),}"> where the cardinality of the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> is the cardinality of an arbitrary basis for this particular <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9175d3b6bb8c26e6acb3fab128b34a294491c876" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.284ex; height:2.676ex;" alt="{\displaystyle L^{2}.}"> If we use complex-valued functions, the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ab400cc4dfd865180cd84c72dc894ca457671f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.458ex; height:2.343ex;" alt="{\displaystyle L^{\infty }}"> is a <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a> <a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a> with pointwise multiplication and conjugation. For many measure spaces, including all sigma-finite ones, it is in fact a commutative <a href="/wiki/Von_Neumann_algebra" title="Von Neumann algebra">von Neumann algebra</a>. An element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ab400cc4dfd865180cd84c72dc894ca457671f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.458ex; height:2.343ex;" alt="{\displaystyle L^{\infty }}"> defines a <a href="/wiki/Bounded_operator" title="Bounded operator">bounded operator</a> on any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"> space by <a href="/wiki/Multiplication_operator" title="Multiplication operator">multiplication</a>. <div class="mw-heading mw-heading3"><h3 id="When_(0_<_p_<_1)">When (0 < p < 1)</h3>[<a href="/w/index.php?title=Lp_space&action=edit&section=10" title="Edit section: When (0 edit</a>]</div> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<p<1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<p<1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73d900254969232ec9def8008e890c03d6d6253b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.338ex; height:2.509ex;" alt="{\displaystyle 0<p<1,}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12496547ebfe44b8cb35e60d89069734ed4154c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.853ex; height:2.843ex;" alt="{\displaystyle L^{p}(\mu )}"> can be defined as above, that is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{p}(f)=\int _{S}|f|^{p}\,d\mu <\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>μ</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{p}(f)=\int _{S}|f|^{p}\,d\mu <\infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d12f3dc1c2a80be92e577b0bf66aaeb461f215" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.789ex; height:5.676ex;" alt="{\displaystyle N_{p}(f)=\int _{S}|f|^{p}\,d\mu <\infty .}"> In this case, however, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-norm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{p}=N_{p}(f)^{1/p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{p}=N_{p}(f)^{1/p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1610a4a7d8d7eb0131f31dd72773eb9c85de23d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.478ex; height:3.509ex;" alt="{\displaystyle \|f\|_{p}=N_{p}(f)^{1/p}}"> does not satisfy the triangle inequality and defines only a <a href="/wiki/Quasi-norm" class="mw-redirect" title="Quasi-norm">quasi-norm</a>. The inequality <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+b)^{p}\leq a^{p}+b^{p},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>≤</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+b)^{p}\leq a^{p}+b^{p},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/177e5a1e2844c128882e3a83034851461ac41ecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.868ex; height:2.843ex;" alt="{\displaystyle (a+b)^{p}\leq a^{p}+b^{p},}"> valid for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\geq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\geq 0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6227e3f20f775a04ae27e075e1114b8d93ba01fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.169ex; height:2.509ex;" alt="{\displaystyle a,b\geq 0,}"> implies that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{p}(f+g)\leq N_{p}(f)+N_{p}(g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo>+</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{p}(f+g)\leq N_{p}(f)+N_{p}(g)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f439617afe1a6722f03935e3880ce5f79880fcbb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.772ex; height:3.009ex;" alt="{\displaystyle N_{p}(f+g)\leq N_{p}(f)+N_{p}(g)}"> and so the function <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{p}(f,g)=N_{p}(f-g)=\|f-g\|_{p}^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo>−</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mo>−</mo> <mi>g</mi> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{p}(f,g)=N_{p}(f-g)=\|f-g\|_{p}^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4b0e9210756d7877464a5e12c4b1cce53fdfa34" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.292ex; height:3.176ex;" alt="{\displaystyle d_{p}(f,g)=N_{p}(f-g)=\|f-g\|_{p}^{p}}"> is a metric on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\mu ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\mu ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e01bcdc24584e0f53bc610e4fa1058c21c2181f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.5ex; height:2.843ex;" alt="{\displaystyle L^{p}(\mu ).}"> The resulting metric space is <a href="/wiki/Complete_metric_space" title="Complete metric space">complete</a>.<a href="#cite_note-FOOTNOTERudin199137-13">[6]</a> In this setting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"> satisfies a reverse Minkowski inequality, that is for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u,v\in L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u,v\in L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eba318b2bb2421c6cfc9d26b95b152a95106e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.974ex; height:2.676ex;" alt="{\displaystyle u,v\in L^{p}}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\Big \|}|u|+|v|{\Big \|}_{p}\geq \|u\|_{p}+\|v\|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo symmetric="true" maxsize="1.623em" minsize="1.623em">‖</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo symmetric="true" maxsize="1.623em" minsize="1.623em">‖</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>≥</mo> <mo fence="false" stretchy="false">‖</mo> <mi>u</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>v</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Big \|}|u|+|v|{\Big \|}_{p}\geq \|u\|_{p}+\|v\|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a84bcb9a5a9d89e3a7c6358fb2b280d4431ebca9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.433ex; height:4.676ex;" alt="{\displaystyle {\Big \|}|u|+|v|{\Big \|}_{p}\geq \|u\|_{p}+\|v\|_{p}}"> This result may be used to prove <a href="/wiki/Clarkson%27s_inequalities" title="Clarkson's inequalities">Clarkson's inequalities</a>, which are in turn used to establish the <a href="/wiki/Uniformly_convex_space" title="Uniformly convex space">uniform convexity</a> of the spaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1<p<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1<p<\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5540fd86346b4798d7447b8de70fe98cf6243d28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.853ex; height:2.509ex;" alt="{\displaystyle 1<p<\infty }"> (<a href="#CITEREFAdamsFournier2003">Adams & Fournier 2003</a>). The space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<p<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<p<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea074f5b36db6eff17f1aa84d73e30e3de12c4d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.691ex; height:2.509ex;" alt="{\displaystyle 0<p<1}"> is an <a href="/wiki/F-space" title="F-space">F-space</a>: it admits a complete translation-invariant metric with respect to which the vector space operations are continuous. It is the prototypical example of an <a href="/wiki/F-space" title="F-space">F-space</a> that, for most reasonable measure spaces, is not <a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">locally convex</a>: in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c352341ab7260ca1a9004da44c897d1395203c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.029ex; height:2.343ex;" alt="{\displaystyle \ell ^{p}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}([0,1]),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}([0,1]),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21d199e66727def8e0ddc2797780054788dfe119" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.751ex; height:2.843ex;" alt="{\displaystyle L^{p}([0,1]),}"> every open convex set containing the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> function is unbounded for the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-quasi-norm; therefore, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> vector does not possess a fundamental system of convex neighborhoods. Specifically, this is true if the measure space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> contains an infinite family of disjoint measurable sets of finite positive measure. The only nonempty convex open set in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}([0,1])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}([0,1])}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d53d55ccc06ce5cc9ec3bede2be3e7933c206ee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.104ex; height:2.843ex;" alt="{\displaystyle L^{p}([0,1])}"> is the entire space. Consequently, there are no nonzero continuous linear functionals on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}([0,1]);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}([0,1]);}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c33199d2ec96bc7ffa77f184aa0760bd5a3c5db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.751ex; height:2.843ex;" alt="{\displaystyle L^{p}([0,1]);}"> the <a href="/wiki/Continuous_dual_space" class="mw-redirect" title="Continuous dual space">continuous dual space</a> is the zero space. In the case of the <a href="/wiki/Counting_measure" title="Counting measure">counting measure</a> on the natural numbers (i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\mu )=\ell ^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\mu )=\ell ^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/207e3a376ce69a0b5cfaf56c22d80d505bd0a9c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.98ex; height:2.843ex;" alt="{\displaystyle L^{p}(\mu )=\ell ^{p}}">), the bounded linear functionals on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c352341ab7260ca1a9004da44c897d1395203c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.029ex; height:2.343ex;" alt="{\displaystyle \ell ^{p}}"> are exactly those that are bounded on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d1156e1c2220628042b0fc51e0c73deb3b7c6d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{1}}">, i.e., those given by sequences in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e024835e5ef8ddeccb2fd0863554f526dfafd5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.492ex; height:2.343ex;" alt="{\displaystyle \ell ^{\infty }.}"> Although <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c352341ab7260ca1a9004da44c897d1395203c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.029ex; height:2.343ex;" alt="{\displaystyle \ell ^{p}}"> does contain non-trivial convex open sets, it fails to have enough of them to give a base for the topology. Having no linear functionals is highly undesirable for the purposes of doing analysis. In case of the Lebesgue measure on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7035fcb9fe3ebecc6bc9f372f82d0352202c8bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{n},}"> rather than work with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<p<1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<p<1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73d900254969232ec9def8008e890c03d6d6253b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.338ex; height:2.509ex;" alt="{\displaystyle 0<p<1,}"> it is common to work with the <a href="/wiki/Hardy_space" title="Hardy space">Hardy space</a> Hp whenever possible, as this has quite a few linear functionals: enough to distinguish points from one another. However, the <a href="/wiki/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach theorem</a> still fails in Hp for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f40fda4df30e9f9f3354c808cc122e5dadbc2d99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p<1}"> (<a href="#CITEREFDuren1970">Duren 1970</a>, §7.5). <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2>[<a href="/w/index.php?title=Lp_space&action=edit&section=11" title="Edit section: Properties">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Hölder's_inequality">Hölder's inequality</h3>[<a href="/w/index.php?title=Lp_space&action=edit&section=12" title="Edit section: Hölder's inequality">edit</a>]</div> Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p,q,r\in [1,\infty ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>,</mo> <mi>r</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p,q,r\in [1,\infty ]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e9c5154a292d43211a6148a4dcb00afe151889e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:14.099ex; height:2.843ex;" alt="{\displaystyle p,q,r\in [1,\infty ]}"> satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}={\tfrac {1}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>q</mi> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}={\tfrac {1}{r}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e67f60890577d74c384e83c8845aaeb72aa3c1cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.918ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}={\tfrac {1}{r}}}">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in L^{p}(S,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in L^{p}(S,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3febfdcc9694e406cc5bfc5788ee30f970706cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.505ex; height:2.843ex;" alt="{\displaystyle f\in L^{p}(S,\mu )}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in L^{q}(S,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\in L^{q}(S,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fdc6e4902bce9780d58fb24eb566ec321e919eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.272ex; height:2.843ex;" alt="{\displaystyle g\in L^{q}(S,\mu )}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle fg\in L^{r}(S,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mi>g</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle fg\in L^{r}(S,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22f2942aadc0eed870eb408eb60b806dcf892b88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.536ex; height:2.843ex;" alt="{\displaystyle fg\in L^{r}(S,\mu )}"> and<a href="#cite_note-FOOTNOTEBahouriCheminDanchin20111–4-14">[7]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|fg\|_{r}~\leq ~\|f\|_{p}\,\|g\|_{q}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mi>g</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mtext> </mtext> <mo>≤</mo> <mtext> </mtext> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">‖</mo> <mi>g</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|fg\|_{r}~\leq ~\|f\|_{p}\,\|g\|_{q}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e801d65fc281dd7014e717f03d6902cf03c2381" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.079ex; height:3.009ex;" alt="{\displaystyle \|fg\|_{r}~\leq ~\|f\|_{p}\,\|g\|_{q}.}"> This inequality, called <a href="/wiki/H%C3%B6lder%27s_inequality" title="Hölder's inequality">Hölder's inequality</a>, is in some sense optimal since if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e6584ba3b7843583b757896c2f0686efc0489e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=1}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is a measurable function such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup _{\|g\|_{q}\leq 1}\,\int _{S}|fg|\,\mathrm {d} \mu ~<~\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">‖</mo> <mi>g</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo>≤</mo> <mn>1</mn> </mrow> </munder> <mspace width="thinmathspace" /> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>μ</mi> <mtext> </mtext> <mo><</mo> <mtext> </mtext> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup _{\|g\|_{q}\leq 1}\,\int _{S}|fg|\,\mathrm {d} \mu ~<~\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3147d04894e329b8a86536c88674299cbef2b49" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.411ex; height:6.676ex;" alt="{\displaystyle \sup _{\|g\|_{q}\leq 1}\,\int _{S}|fg|\,\mathrm {d} \mu ~<~\infty }"> where the <a href="/wiki/Supremum" class="mw-redirect" title="Supremum">supremum</a> is taken over the closed unit ball of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{q}(S,\mu ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{q}(S,\mu ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2725d959d7b5a60894ddaa0c65e50e4c6974d37c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.962ex; height:2.843ex;" alt="{\displaystyle L^{q}(S,\mu ),}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in L^{p}(S,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in L^{p}(S,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3febfdcc9694e406cc5bfc5788ee30f970706cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.505ex; height:2.843ex;" alt="{\displaystyle f\in L^{p}(S,\mu )}"> and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{p}~=~\sup _{\|g\|_{q}\leq 1}\,\int _{S}fg\,\mathrm {d} \mu .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">‖</mo> <mi>g</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo>≤</mo> <mn>1</mn> </mrow> </munder> <mspace width="thinmathspace" /> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mi>f</mi> <mi>g</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>μ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{p}~=~\sup _{\|g\|_{q}\leq 1}\,\int _{S}fg\,\mathrm {d} \mu .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c408c7d2d1e638ca5c50734c11d5deea04f1015" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.49ex; height:6.676ex;" alt="{\displaystyle \|f\|_{p}~=~\sup _{\|g\|_{q}\leq 1}\,\int _{S}fg\,\mathrm {d} \mu .}"> <div class="mw-heading mw-heading3"><h3 id="Generalized_Minkowski_inequality">Generalized Minkowski inequality</h3>[<a href="/w/index.php?title=Lp_space&action=edit&section=13" title="Edit section: Generalized Minkowski inequality">edit</a>]</div> <a href="/wiki/Minkowski_inequality" title="Minkowski inequality">Minkowski inequality</a>, which states that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44e951f6e15fdb1bdd7d223bbbe628955251a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.063ex; height:3.009ex;" alt="{\displaystyle \|\cdot \|_{p}}"> satisfies the <a href="/wiki/Triangle_inequality" title="Triangle inequality">triangle inequality</a>, can be generalized: If the measurable function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:M\times N\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>:</mo> <mi>M</mi> <mo>×</mo> <mi>N</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F:M\times N\to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aedb9aca25fb1593d254abdc5573a7d6452ff93a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.316ex; height:2.176ex;" alt="{\displaystyle F:M\times N\to \mathbb {R} }"> is non-negative (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/395032f1e542cd44f626bf41accccb80b1f83d19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.687ex; height:2.843ex;" alt="{\displaystyle (M,\mu )}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (N,\nu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (N,\nu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5006d3e48f28ebdd5c604e227affb9d13a9d1f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.139ex; height:2.843ex;" alt="{\displaystyle (N,\nu )}"> are measure spaces) then for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p\leq q\leq \infty ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi>q</mi> <mo>≤</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq p\leq q\leq \infty ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb0ff03b8517588c498d458c115116c6f7dcfcf9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.667ex; height:2.509ex;" alt="{\displaystyle 1\leq p\leq q\leq \infty ,}"><a href="#cite_note-FOOTNOTEBahouriCheminDanchin20114-15">[8]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\|\left\|F(\,\cdot ,n)\right\|_{L^{p}(M,\mu )}\right\|_{L^{q}(N,\nu )}~\leq ~\left\|\left\|F(m,\cdot )\right\|_{L^{q}(N,\nu )}\right\|_{L^{p}(M,\mu )}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo symmetric="true">‖</mo> <msub> <mrow> <mo symmetric="true">‖</mo> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mspace width="thinmathspace" /> <mo>⋅</mo> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mtext> </mtext> <mo>≤</mo> <mtext> </mtext> <msub> <mrow> <mo symmetric="true">‖</mo> <msub> <mrow> <mo symmetric="true">‖</mo> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\|\left\|F(\,\cdot ,n)\right\|_{L^{p}(M,\mu )}\right\|_{L^{q}(N,\nu )}~\leq ~\left\|\left\|F(m,\cdot )\right\|_{L^{q}(N,\nu )}\right\|_{L^{p}(M,\mu )}\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e4aa49cb76d3d0f791b38822b5a7e680852171" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:54.748ex; height:4.343ex;" alt="{\displaystyle \left\|\left\|F(\,\cdot ,n)\right\|_{L^{p}(M,\mu )}\right\|_{L^{q}(N,\nu )}~\leq ~\left\|\left\|F(m,\cdot )\right\|_{L^{q}(N,\nu )}\right\|_{L^{p}(M,\mu )}\ .}"> <div class="mw-heading mw-heading3"><h3 id="Atomic_decomposition">Atomic decomposition</h3>[<a href="/w/index.php?title=Lp_space&action=edit&section=14" title="Edit section: Atomic decomposition">edit</a>]</div> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq p<\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d42661a6d23dcde0fb7ae1b611e233958d13da9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.853ex; height:2.509ex;" alt="{\displaystyle 1\leq p<\infty }"> then every non-negative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in L^{p}(\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in L^{p}(\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb03acc3f191b3c787dc1aad2b77515de09b5c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.972ex; height:2.843ex;" alt="{\displaystyle f\in L^{p}(\mu )}"> has an atomic decomposition,<a href="#cite_note-FOOTNOTEBahouriCheminDanchin20117–8-16">[9]</a> meaning that there exist a sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r_{n})_{n\in \mathbb {Z} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r_{n})_{n\in \mathbb {Z} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6160ca32b2207888472b8cd843a01fb951600964" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.487ex; height:2.843ex;" alt="{\displaystyle (r_{n})_{n\in \mathbb {Z} }}"> of non-negative real numbers and a sequence of non-negative functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{n})_{n\in \mathbb {Z} },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f_{n})_{n\in \mathbb {Z} },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4493c3171226af5ea889ec3008ebc235c427c71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.225ex; height:2.843ex;" alt="{\displaystyle (f_{n})_{n\in \mathbb {Z} },}"> called the atoms, whose supports <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\operatorname {supp} f_{n}\right)_{n\in \mathbb {Z} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <mrow> <mi>supp</mi> <mo>⁡</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\operatorname {supp} f_{n}\right)_{n\in \mathbb {Z} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7179e9e9f3f68ba58515c5d897e9441b96462bfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.759ex; height:3.009ex;" alt="{\displaystyle \left(\operatorname {supp} f_{n}\right)_{n\in \mathbb {Z} }}"> are <a href="/wiki/Disjoint_sets" title="Disjoint sets">pairwise disjoint sets</a> of measure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \left(\operatorname {supp} f_{n}\right)\leq 2^{n+1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mrow> <mo>(</mo> <mrow> <mi>supp</mi> <mo>⁡</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>≤</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \left(\operatorname {supp} f_{n}\right)\leq 2^{n+1},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64500fbb8695e64010367fe376164c7f8aa5d7ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.363ex; height:3.176ex;" alt="{\displaystyle \mu \left(\operatorname {supp} f_{n}\right)\leq 2^{n+1},}"> such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f~=~\sum _{n\in \mathbb {Z} }r_{n}\,f_{n}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f~=~\sum _{n\in \mathbb {Z} }r_{n}\,f_{n}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8aae45cb917926c062b879b04ee72e9ef908598" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:15.713ex; height:5.676ex;" alt="{\displaystyle f~=~\sum _{n\in \mathbb {Z} }r_{n}\,f_{n}\,,}"> and for every integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {Z} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {Z} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32f0d0d4c10d0cdc803b261e36c2a7c0f3c122e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.433ex; height:2.509ex;" alt="{\displaystyle n\in \mathbb {Z} ,}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f_{n}\|_{\infty }~\leq ~2^{-{\tfrac {n}{p}}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mtext> </mtext> <mo>≤</mo> <mtext> </mtext> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mi>p</mi> </mfrac> </mstyle> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f_{n}\|_{\infty }~\leq ~2^{-{\tfrac {n}{p}}}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46a6de16710786c0a63d3f33177d57d139115e29" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.347ex; height:4.176ex;" alt="{\displaystyle \|f_{n}\|_{\infty }~\leq ~2^{-{\tfrac {n}{p}}}\,,}"> and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}\|f\|_{p}^{p}~\leq ~\sum _{n\in \mathbb {Z} }r_{n}^{p}~\leq ~2\|f\|_{p}^{p}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <mtext> </mtext> <mo>≤</mo> <mtext> </mtext> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <mtext> </mtext> <mo>≤</mo> <mtext> </mtext> <mn>2</mn> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}\|f\|_{p}^{p}~\leq ~\sum _{n\in \mathbb {Z} }r_{n}^{p}~\leq ~2\|f\|_{p}^{p}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23a81c249d5f54891b0a88744880757a40e78813" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.096ex; height:5.676ex;" alt="{\displaystyle {\tfrac {1}{2}}\|f\|_{p}^{p}~\leq ~\sum _{n\in \mathbb {Z} }r_{n}^{p}~\leq ~2\|f\|_{p}^{p}\,,}"> and where moreover, the sequence of functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r_{n}f_{n})_{n\in \mathbb {Z} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r_{n}f_{n})_{n\in \mathbb {Z} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60e77a01120a61d50994d79231053d7832defde6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.845ex; height:2.843ex;" alt="{\displaystyle (r_{n}f_{n})_{n\in \mathbb {Z} }}"> depends only on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> (it is independent of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">).<a href="#cite_note-FOOTNOTEBahouriCheminDanchin20117–8-16">[9]</a> These inequalities guarantee that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f_{n}\|_{p}^{p}\leq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <mo>≤</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f_{n}\|_{p}^{p}\leq 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c45c84edcb464fd241dc06261e36d23df5f2ab8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.003ex; height:3.176ex;" alt="{\displaystyle \|f_{n}\|_{p}^{p}\leq 2}"> for all integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> while the supports of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{n})_{n\in \mathbb {Z} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f_{n})_{n\in \mathbb {Z} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/542f7d07a04bacfe63f9ffa3c5327cb2a63c54c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.578ex; height:2.843ex;" alt="{\displaystyle (f_{n})_{n\in \mathbb {Z} }}"> being pairwise disjoint implies<a href="#cite_note-FOOTNOTEBahouriCheminDanchin20117–8-16">[9]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{p}^{p}~=~\sum _{n\in \mathbb {Z} }r_{n}^{p}\,\|f_{n}\|_{p}^{p}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{p}^{p}~=~\sum _{n\in \mathbb {Z} }r_{n}^{p}\,\|f_{n}\|_{p}^{p}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bae507a64c60542d306486b8a1867d5e2ef4049" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.482ex; height:5.676ex;" alt="{\displaystyle \|f\|_{p}^{p}~=~\sum _{n\in \mathbb {Z} }r_{n}^{p}\,\|f_{n}\|_{p}^{p}\,.}"> An atomic decomposition can be explicitly given by first defining for every integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {Z} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {Z} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32f0d0d4c10d0cdc803b261e36c2a7c0f3c122e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.433ex; height:2.509ex;" alt="{\displaystyle n\in \mathbb {Z} ,}"><a href="#cite_note-FOOTNOTEBahouriCheminDanchin20117–8-16">[9]</a><a href="#cite_note-17">[note 7]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n}=\inf\{t\in \mathbb {R} :\mu (f>t)<2^{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">inf</mo> <mo fence="false" stretchy="false">{</mo> <mi>t</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>:</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>></mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo><</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{n}=\inf\{t\in \mathbb {R} :\mu (f>t)<2^{n}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/914b3cd736e3ef0b646b65937daf271db3d3b695" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.335ex; height:2.843ex;" alt="{\displaystyle t_{n}=\inf\{t\in \mathbb {R} :\mu (f>t)<2^{n}\}}"> and then letting <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{n}~=~2^{n/p}\,t_{n}~{\text{ and }}\quad f_{n}~=~{\frac {f}{r_{n}}}\,\mathbf {1} _{(t_{n+1}<f\leq t_{n})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mspace width="1em" /> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>f</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo><</mo> <mi>f</mi> <mo>≤</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n}~=~2^{n/p}\,t_{n}~{\text{ and }}\quad f_{n}~=~{\frac {f}{r_{n}}}\,\mathbf {1} _{(t_{n+1}<f\leq t_{n})}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b828376298914eeb939b86b6bc8a1b5f1844eab" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:42.054ex; height:5.676ex;" alt="{\displaystyle r_{n}~=~2^{n/p}\,t_{n}~{\text{ and }}\quad f_{n}~=~{\frac {f}{r_{n}}}\,\mathbf {1} _{(t_{n+1}<f\leq t_{n})}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (f>t)=\mu (\{s:f(s)>t\})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>></mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mi>s</mi> <mo>:</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mi>t</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (f>t)=\mu (\{s:f(s)>t\})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecbd7cb97ded44af8e0ef327e6cd40da0015b0d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.206ex; height:2.843ex;" alt="{\displaystyle \mu (f>t)=\mu (\{s:f(s)>t\})}"> denotes the measure of the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f>t):=\{s\in S:f(s)>t\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>></mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>s</mi> <mo>∈</mo> <mi>S</mi> <mo>:</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>></mo> <mi>t</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f>t):=\{s\in S:f(s)>t\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1634cb9b76d614ba5c089e8000e8a44d29871cfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.58ex; height:2.843ex;" alt="{\displaystyle (f>t):=\{s\in S:f(s)>t\}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} _{(t_{n+1}<f\leq t_{n})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo><</mo> <mi>f</mi> <mo>≤</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{(t_{n+1}<f\leq t_{n})}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c134bd7740a3473248b95df2a5c40d0e32f8ae02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.132ex; height:3.009ex;" alt="{\displaystyle \mathbf {1} _{(t_{n+1}<f\leq t_{n})}}"> denotes the <a href="/wiki/Indicator_function" title="Indicator function">indicator function</a> of the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t_{n+1}<f\leq t_{n}):=\{s\in S:t_{n+1}<f(s)\leq t_{n}\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo><</mo> <mi>f</mi> <mo>≤</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>s</mi> <mo>∈</mo> <mi>S</mi> <mo>:</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo><</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t_{n+1}<f\leq t_{n}):=\{s\in S:t_{n+1}<f(s)\leq t_{n}\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4a063e69afbe39974e6e35d4dc7a3705b48b44c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.178ex; height:2.843ex;" alt="{\displaystyle (t_{n+1}<f\leq t_{n}):=\{s\in S:t_{n+1}<f(s)\leq t_{n}\}.}"> The sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t_{n})_{n\in \mathbb {Z} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t_{n})_{n\in \mathbb {Z} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79d5147e962b4da3ddf8c37884dfff96a990bf1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.278ex; height:2.843ex;" alt="{\displaystyle (t_{n})_{n\in \mathbb {Z} }}"> is decreasing and converges to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\to \infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\to \infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8a34a9f62668de90200a6cbde865c27af2cdbb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.979ex; height:1.843ex;" alt="{\displaystyle n\to \infty .}"><a href="#cite_note-FOOTNOTEBahouriCheminDanchin20117–8-16">[9]</a> Consequently, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{n}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72864eb179eae8b3031b64c76d0254487c2bd8ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.319ex; height:2.509ex;" alt="{\displaystyle t_{n}=0}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n+1}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{n+1}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99e6cba959acb3b04a513147707fd50c81c68a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.42ex; height:2.509ex;" alt="{\displaystyle t_{n+1}=0}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t_{n+1}<f\leq t_{n})=\varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo><</mo> <mi>f</mi> <mo>≤</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t_{n+1}<f\leq t_{n})=\varnothing }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6807bf080a48fc297e3b750675a72715d5330a18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.408ex; height:2.843ex;" alt="{\displaystyle (t_{n+1}<f\leq t_{n})=\varnothing }"> so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}={\frac {1}{r_{n}}}\,f\,\mathbf {1} _{(t_{n+1}<f\leq t_{n})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>f</mi> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo><</mo> <mi>f</mi> <mo>≤</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{n}={\frac {1}{r_{n}}}\,f\,\mathbf {1} _{(t_{n+1}<f\leq t_{n})}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b197c30809c60093dc614b7f28b2b0d682c39b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.744ex; height:5.509ex;" alt="{\displaystyle f_{n}={\frac {1}{r_{n}}}\,f\,\mathbf {1} _{(t_{n+1}<f\leq t_{n})}}"> is identically equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> (in particular, the division <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{r_{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{r_{n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a378074c9b0f22ab219a343e23aa027f23d2d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.543ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{r_{n}}}}"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{n}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eed0d09b179ca51655c2af9e6b4d9a02779a58b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.528ex; height:2.509ex;" alt="{\displaystyle r_{n}=0}"> causes no issues). The <a href="/wiki/Complementary_cumulative_distribution_function" class="mw-redirect" title="Complementary cumulative distribution function">complementary cumulative distribution function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in \mathbb {R} \mapsto \mu (|f|>t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">↦</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in \mathbb {R} \mapsto \mu (|f|>t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d440d592e0bbf85a992d496f281d510e59410ed9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.694ex; height:2.843ex;" alt="{\displaystyle t\in \mathbb {R} \mapsto \mu (|f|>t)}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f|=f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f|=f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc5d5afd4f026a15298fd4ec97dcec2743ba3bf5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.949ex; height:2.843ex;" alt="{\displaystyle |f|=f}"> that was used to define the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/271566db7e8ca8616a4dc3efb6c5982a2d987ee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.058ex; height:2.343ex;" alt="{\displaystyle t_{n}}"> also appears in the definition of the weak <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}">-norm (given below) and can be used to express the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-norm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44e951f6e15fdb1bdd7d223bbbe628955251a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.063ex; height:3.009ex;" alt="{\displaystyle \|\cdot \|_{p}}"> (for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq p<\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d42661a6d23dcde0fb7ae1b611e233958d13da9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.853ex; height:2.509ex;" alt="{\displaystyle 1\leq p<\infty }">) of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in L^{p}(S,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in L^{p}(S,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3febfdcc9694e406cc5bfc5788ee30f970706cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.505ex; height:2.843ex;" alt="{\displaystyle f\in L^{p}(S,\mu )}"> as the integral<a href="#cite_note-FOOTNOTEBahouriCheminDanchin20117–8-16">[9]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{p}^{p}~=~p\,\int _{0}^{\infty }t^{p-1}\mu (|f|>t)\,\mathrm {d} t\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mi>p</mi> <mspace width="thinmathspace" /> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>μ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{p}^{p}~=~p\,\int _{0}^{\infty }t^{p-1}\mu (|f|>t)\,\mathrm {d} t\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/220f0963e687bbb529c88ba4a54585c53eb6dca7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.866ex; height:5.843ex;" alt="{\displaystyle \|f\|_{p}^{p}~=~p\,\int _{0}^{\infty }t^{p-1}\mu (|f|>t)\,\mathrm {d} t\,,}"> where the integration is with respect to the usual Lebesgue measure on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,\infty ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,\infty ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59160f5aaeb8302db9510e8026131f38498c3937" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.976ex; height:2.843ex;" alt="{\displaystyle (0,\infty ).}"> <div class="mw-heading mw-heading3"><h3 id="Dual_spaces">Dual spaces</h3>[<a href="/w/index.php?title=Lp_space&action=edit&section=15" title="Edit section: Dual spaces">edit</a>]</div> The <a href="/wiki/Continuous_dual" class="mw-redirect" title="Continuous dual">dual space</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12496547ebfe44b8cb35e60d89069734ed4154c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.853ex; height:2.843ex;" alt="{\displaystyle L^{p}(\mu )}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1<p<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1<p<\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5540fd86346b4798d7447b8de70fe98cf6243d28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.853ex; height:2.509ex;" alt="{\displaystyle 1<p<\infty }"> has a natural isomorphism with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{q}(\mu ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{q}(\mu ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65943f5156050bd256d2f4b900c5922ee7152a21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.429ex; height:2.843ex;" alt="{\displaystyle L^{q}(\mu ),}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"> is such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>q</mi> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0258202dfc8069365da867d2ef863a2a53f04ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.422ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}">. This isomorphism associates <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in L^{q}(\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\in L^{q}(\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac6d04ed0823bca8efab9efb3a457537d0aeb0a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.739ex; height:2.843ex;" alt="{\displaystyle g\in L^{q}(\mu )}"> with the functional <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa _{p}(g)\in L^{p}(\mu )^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa _{p}(g)\in L^{p}(\mu )^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/668355e61e91adff339b96028f5a10648a805bff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.071ex; height:3.009ex;" alt="{\displaystyle \kappa _{p}(g)\in L^{p}(\mu )^{*}}"> defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\mapsto \kappa _{p}(g)(f)=\int fg\,\mathrm {d} \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">↦</mo> <msub> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <mi>f</mi> <mi>g</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\mapsto \kappa _{p}(g)(f)=\int fg\,\mathrm {d} \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60b26f2f1d45ff3c33e9bebdec6aef149202e06d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.459ex; height:5.676ex;" alt="{\displaystyle f\mapsto \kappa _{p}(g)(f)=\int fg\,\mathrm {d} \mu }"> for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in L^{p}(\mu ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in L^{p}(\mu ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7ed08cc045c705e89d9f0d16bb62e1a8c93e8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.619ex; height:2.843ex;" alt="{\displaystyle f\in L^{p}(\mu ).}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa _{p}:L^{q}(\mu )\to L^{p}(\mu )^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>:</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa _{p}:L^{q}(\mu )\to L^{p}(\mu )^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2e7adac80fa7eba3c8f8a82c4780072897a3c67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.639ex; height:3.009ex;" alt="{\displaystyle \kappa _{p}:L^{q}(\mu )\to L^{p}(\mu )^{*}}"> is a well defined continuous linear mapping which is an <a href="/wiki/Isometry" title="Isometry">isometry</a> by the <a href="/wiki/H%C3%B6lder%27s_inequality#Extremal_equality" title="Hölder's inequality">extremal case</a> of Hölder's inequality. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S,\Sigma ,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S,\Sigma ,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cf1dfd7f18980638e424a78b4f5fe40df597c8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.456ex; height:2.843ex;" alt="{\displaystyle (S,\Sigma ,\mu )}"> is a <a href="/wiki/Measure_space#Important_classes_of_measure_spaces" title="Measure space"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }">-finite measure space</a> one can use the <a href="/wiki/Radon%E2%80%93Nikodym_theorem" title="Radon–Nikodym theorem">Radon–Nikodym theorem</a> to show that any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\in L^{p}(\mu )^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\in L^{p}(\mu )^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f2f926de5a351c048984bc6c3dc363ab93bdb47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.575ex; height:2.843ex;" alt="{\displaystyle G\in L^{p}(\mu )^{*}}"> can be expressed this way, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dca34d1a0228682daff98ed2767294435e07cb4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.398ex; height:2.343ex;" alt="{\displaystyle \kappa _{p}}"> is an <a href="/wiki/Isometry#Definition" title="Isometry">isometric isomorphism</a> of <a href="/wiki/Banach_space" title="Banach space">Banach spaces</a>.<a href="#cite_note-FOOTNOTERudin1987Theorem_6.16-18">[10]</a> Hence, it is usual to say simply that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{q}(\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{q}(\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9568c4262b3e1c017b6f80111b7e5b3d1a0e485c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.782ex; height:2.843ex;" alt="{\displaystyle L^{q}(\mu )}"> is the <a href="/wiki/Continuous_dual_space" class="mw-redirect" title="Continuous dual space">continuous dual space</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\mu ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\mu ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e01bcdc24584e0f53bc610e4fa1058c21c2181f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.5ex; height:2.843ex;" alt="{\displaystyle L^{p}(\mu ).}"> For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1<p<\infty ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1<p<\infty ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/203e39618106a0a92baad7c6443a390d1dec4792" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.499ex; height:2.509ex;" alt="{\displaystyle 1<p<\infty ,}"> the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12496547ebfe44b8cb35e60d89069734ed4154c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.853ex; height:2.843ex;" alt="{\displaystyle L^{p}(\mu )}"> is <a href="/wiki/Reflexive_space" title="Reflexive space">reflexive</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dca34d1a0228682daff98ed2767294435e07cb4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.398ex; height:2.343ex;" alt="{\displaystyle \kappa _{p}}"> be as above and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa _{q}:L^{p}(\mu )\to L^{q}(\mu )^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo>:</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa _{q}:L^{p}(\mu )\to L^{q}(\mu )^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b8e76dbb2bf3f34a9e7af76317993bddd4bc434" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.568ex; height:3.009ex;" alt="{\displaystyle \kappa _{q}:L^{p}(\mu )\to L^{q}(\mu )^{*}}"> be the corresponding linear isometry. Consider the map from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12496547ebfe44b8cb35e60d89069734ed4154c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.853ex; height:2.843ex;" alt="{\displaystyle L^{p}(\mu )}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\mu )^{**},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> <mo>∗</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\mu )^{**},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d74925f21631055fdf4430d8b70d7cf4a4c855f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.376ex; height:2.843ex;" alt="{\displaystyle L^{p}(\mu )^{**},}"> obtained by composing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa _{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa _{q}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e650eaa361bfbafc4c9497577fa603b852aeb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.328ex; height:2.343ex;" alt="{\displaystyle \kappa _{q}}"> with the <a href="/wiki/Dual_space#Transpose_of_a_continuous_linear_map" title="Dual space">transpose</a> (or adjoint) of the inverse of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa _{p}:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa _{p}:}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57f9eaebab7f6e4af1ca19b8803994817bed5600" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.69ex; height:2.343ex;" alt="{\displaystyle \kappa _{p}:}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{p}:L^{p}(\mu )\mathrel {\overset {\kappa _{q}}{\longrightarrow }} L^{q}(\mu )^{*}\mathrel {\overset {\left(\kappa _{p}^{-1}\right)^{*}}{\longrightarrow }} L^{p}(\mu )^{**}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>:</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-REL"> <mover> <mo stretchy="false">⟶</mo> <msub> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mover> </mrow> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-REL"> <mover> <mo stretchy="false">⟶</mo> <msup> <mrow> <mo>(</mo> <msubsup> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mover> </mrow> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j_{p}:L^{p}(\mu )\mathrel {\overset {\kappa _{q}}{\longrightarrow }} L^{q}(\mu )^{*}\mathrel {\overset {\left(\kappa _{p}^{-1}\right)^{*}}{\longrightarrow }} L^{p}(\mu )^{**}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03ad0cec951343aa151a840744a35f9cfb004a83" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.027ex; width:36.397ex; height:6.009ex;" alt="{\displaystyle j_{p}:L^{p}(\mu )\mathrel {\overset {\kappa _{q}}{\longrightarrow }} L^{q}(\mu )^{*}\mathrel {\overset {\left(\kappa _{p}^{-1}\right)^{*}}{\longrightarrow }} L^{p}(\mu )^{**}}"> This map coincides with the <a href="/wiki/Reflexive_space#Definitions" title="Reflexive space">canonical embedding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12496547ebfe44b8cb35e60d89069734ed4154c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.853ex; height:2.843ex;" alt="{\displaystyle L^{p}(\mu )}"> into its bidual. Moreover, the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e1c8c575a1a35c4afbaf156d8fe96410aeb8dce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.027ex; width:2.044ex; height:2.843ex;" alt="{\displaystyle j_{p}}"> is onto, as composition of two onto isometries, and this proves reflexivity. If the measure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> is <a href="/wiki/Sigma-finite" class="mw-redirect" title="Sigma-finite">sigma-finite</a>, then the dual of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}(\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{1}(\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b81a45f5ed33188fc3e64cd079882bbc7de34c24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.848ex; height:3.176ex;" alt="{\displaystyle L^{1}(\mu )}"> is isometrically isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty }(\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }(\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b7867eb72dc22e91568af1af857fd364f42458c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.669ex; height:2.843ex;" alt="{\displaystyle L^{\infty }(\mu )}"> (more precisely, the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa _{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78471d428eed0c2e016ad82fb2f30af4cedc86eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.009ex;" alt="{\displaystyle \kappa _{1}}"> corresponding to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c29a2f2fb3f642618036ed7a79712202e7ada924" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=1}"> is an isometry from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty }(\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }(\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b7867eb72dc22e91568af1af857fd364f42458c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.669ex; height:2.843ex;" alt="{\displaystyle L^{\infty }(\mu )}"> onto <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}(\mu )^{*}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{1}(\mu )^{*}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac5ef94f53b009619cb92ed69ddc364b3d4707a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.549ex; height:3.176ex;" alt="{\displaystyle L^{1}(\mu )^{*}.}"> The dual of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty }(\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }(\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b7867eb72dc22e91568af1af857fd364f42458c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.669ex; height:2.843ex;" alt="{\displaystyle L^{\infty }(\mu )}"> is subtler. Elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty }(\mu )^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }(\mu )^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/366e4f5983b27d35c77f0cbe3cb0cc732fd6e454" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.723ex; height:2.843ex;" alt="{\displaystyle L^{\infty }(\mu )^{*}}"> can be identified with bounded signed finitely additive measures on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> that are <a href="/wiki/Absolutely_continuous" class="mw-redirect" title="Absolutely continuous">absolutely continuous</a> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1ef6db045c1f6193799bd25a4b68ba9f78646d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.049ex; height:2.176ex;" alt="{\displaystyle \mu .}"> See <a href="/wiki/Ba_space" title="Ba space">ba space</a> for more details. If we assume the axiom of choice, this space is much bigger than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}(\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{1}(\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b81a45f5ed33188fc3e64cd079882bbc7de34c24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.848ex; height:3.176ex;" alt="{\displaystyle L^{1}(\mu )}"> except in some trivial cases. However, <a href="/wiki/Saharon_Shelah" title="Saharon Shelah">Saharon Shelah</a> proved that there are relatively consistent extensions of <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a> (ZF + <a href="/wiki/Axiom_of_dependent_choice" title="Axiom of dependent choice">DC</a> + "Every subset of the real numbers has the <a href="/wiki/Baire_property" class="mw-redirect" title="Baire property">Baire property</a>") in which the dual of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8348195cf09473662c6f59e6717722a6fc01d0f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.845ex; height:2.343ex;" alt="{\displaystyle \ell ^{\infty }}"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/615f8bbfccb6efabd743f484610f32c6898a5a98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.671ex; height:2.676ex;" alt="{\displaystyle \ell ^{1}.}"><a href="#cite_note-19">[11]</a> <div class="mw-heading mw-heading3"><h3 id="Embeddings">Embeddings</h3>[<a href="/w/index.php?title=Lp_space&action=edit&section=16" title="Edit section: Embeddings">edit</a>]</div> Colloquially, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p<q\leq \infty ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>q</mi> <mo>≤</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq p<q\leq \infty ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61cfd0e9af2913201c6b763dda6f381423411cde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.667ex; height:2.509ex;" alt="{\displaystyle 1\leq p<q\leq \infty ,}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(S,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(S,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d5a4c54613c1c773a1caa81a731c340f546e320" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.386ex; height:2.843ex;" alt="{\displaystyle L^{p}(S,\mu )}"> contains functions that are more locally singular, while elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{q}(S,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{q}(S,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/182c494465bba1a916d33599db7886355dbce203" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.316ex; height:2.843ex;" alt="{\displaystyle L^{q}(S,\mu )}"> can be more spread out. Consider the <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a> on the half line <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,\infty ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,\infty ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59160f5aaeb8302db9510e8026131f38498c3937" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.976ex; height:2.843ex;" alt="{\displaystyle (0,\infty ).}"> A continuous function in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74c288d1089f1ec85b01b4de25c441fc792bd2d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{1}}"> might blow up near <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> but must decay sufficiently fast toward infinity. On the other hand, continuous functions in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ab400cc4dfd865180cd84c72dc894ca457671f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.458ex; height:2.343ex;" alt="{\displaystyle L^{\infty }}"> need not decay at all but no blow-up is allowed. More formally, suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<p<q\leq \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>q</mi> <mo>≤</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<p<q\leq \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d48d1f2d68072a17daf8821c5d2feeb86bdee8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.021ex; height:2.509ex;" alt="{\displaystyle 0<p<q\leq \infty }">, then:<a href="#cite_note-VillaniEmbeddings-20">[12]</a> <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{q}(S,\mu )\subseteq L^{p}(S,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>⊆</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{q}(S,\mu )\subseteq L^{p}(S,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dda1471623f2c41ebbbe8c427750801820e6af3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.8ex; height:2.843ex;" alt="{\displaystyle L^{q}(S,\mu )\subseteq L^{p}(S,\mu )}"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> does not contain sets of finite but arbitrarily large measure (e.g. any <a href="/wiki/Finite_measure" title="Finite measure">finite measure</a>).</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(S,\mu )\subseteq L^{q}(S,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>⊆</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(S,\mu )\subseteq L^{q}(S,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/022188d8743e76773c50b45a0ed635dfdc1585d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.8ex; height:2.843ex;" alt="{\displaystyle L^{p}(S,\mu )\subseteq L^{q}(S,\mu )}"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> does not contain sets of non-zero but arbitrarily small measure (e.g. the <a href="/wiki/Counting_measure" title="Counting measure">counting measure</a>).</li></ol> Neither condition holds for the Lebesgue measure on the real line while both conditions holds for the <a href="/wiki/Counting_measure" title="Counting measure">counting measure</a> on any finite set. As a consequence of the <a href="/wiki/Closed_graph_theorem" title="Closed graph theorem">closed graph theorem</a>, the embedding is continuous, i.e., the <a href="/wiki/Identity_operator" class="mw-redirect" title="Identity operator">identity operator</a> is a bounded linear map from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{q}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f58b2af9047d1bd68a9937af5b427b3fa4ab0a44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.571ex; height:2.343ex;" alt="{\displaystyle L^{q}}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"> in the first case and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{q}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f58b2af9047d1bd68a9937af5b427b3fa4ab0a44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.571ex; height:2.343ex;" alt="{\displaystyle L^{q}}"> in the second. Indeed, if the domain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> has finite measure, one can make the following explicit calculation using <a href="/wiki/H%C3%B6lder%27s_inequality" title="Hölder's inequality">Hölder's inequality</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \|\mathbf {1} f^{p}\|_{1}\leq \|\mathbf {1} \|_{q/(q-p)}\|f^{p}\|_{q/p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \|\mathbf {1} f^{p}\|_{1}\leq \|\mathbf {1} \|_{q/(q-p)}\|f^{p}\|_{q/p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4ff1bd311327bfa2d6d69d14433776d59e888e2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:27.73ex; height:3.176ex;" alt="{\displaystyle \ \|\mathbf {1} f^{p}\|_{1}\leq \|\mathbf {1} \|_{q/(q-p)}\|f^{p}\|_{q/p}}"> leading to <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \|f\|_{p}\leq \mu (S)^{1/p-1/q}\|f\|_{q}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>≤</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>S</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> </mrow> </msup> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \|f\|_{p}\leq \mu (S)^{1/p-1/q}\|f\|_{q}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4eeaef2827bb6faf8f1e9f34b34046498cd48ec" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.673ex; height:3.509ex;" alt="{\displaystyle \ \|f\|_{p}\leq \mu (S)^{1/p-1/q}\|f\|_{q}.}"> The constant appearing in the above inequality is optimal, in the sense that the <a href="/wiki/Operator_norm" title="Operator norm">operator norm</a> of the identity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I:L^{q}(S,\mu )\to L^{p}(S,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>:</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I:L^{q}(S,\mu )\to L^{p}(S,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1018ba1a9f74c601fa77ebc5db47856d5379aa2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.425ex; height:2.843ex;" alt="{\displaystyle I:L^{q}(S,\mu )\to L^{p}(S,\mu )}"> is precisely <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|I\|_{q,p}=\mu (S)^{1/p-1/q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>I</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo>,</mo> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>S</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|I\|_{q,p}=\mu (S)^{1/p-1/q}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a3ba5e9b671eb652c8d8544a68e29878c966fa5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.96ex; height:3.509ex;" alt="{\displaystyle \|I\|_{q,p}=\mu (S)^{1/p-1/q}}"> the case of equality being achieved exactly when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60eab5895ca5fe74555e9d59e499c94e1d2ef3f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.509ex;" alt="{\displaystyle f=1}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }">-almost-everywhere. <div class="mw-heading mw-heading3"><h3 id="Dense_subspaces">Dense subspaces</h3>[<a href="/w/index.php?title=Lp_space&action=edit&section=17" title="Edit section: Dense subspaces">edit</a>]</div> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq p<\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d42661a6d23dcde0fb7ae1b611e233958d13da9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.853ex; height:2.509ex;" alt="{\displaystyle 1\leq p<\infty }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S,\Sigma ,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S,\Sigma ,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cf1dfd7f18980638e424a78b4f5fe40df597c8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.456ex; height:2.843ex;" alt="{\displaystyle (S,\Sigma ,\mu )}"> be a measure space and consider an integrable <a href="/wiki/Simple_function" title="Simple function">simple function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=\sum _{j=1}^{n}a_{j}\mathbf {1} _{A_{j}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=\sum _{j=1}^{n}a_{j}\mathbf {1} _{A_{j}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/135a4ef2e73a9ebabb863d0eaf6c1b74cb920dd6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:14.421ex; height:7.176ex;" alt="{\displaystyle f=\sum _{j=1}^{n}a_{j}\mathbf {1} _{A_{j}},}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0096fb78d6843c9fb67a840dc796b61ad93eec2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.14ex; height:2.343ex;" alt="{\displaystyle a_{j}}"> are scalars, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{j}\in \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∈</mo> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{j}\in \Sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4c261d943a44f56fb1c3b9598fb9a89a4511d58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.172ex; height:2.843ex;" alt="{\displaystyle A_{j}\in \Sigma }"> has finite measure and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {1} }_{A_{j}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {1} }_{A_{j}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e12b075d31cd38692e63c20ade8ab8a155ea05b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.516ex; height:3.009ex;" alt="{\displaystyle {\mathbf {1} }_{A_{j}}}"> is the <a href="/wiki/Indicator_function" title="Indicator function">indicator function</a> of the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{j},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{j},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c58c735a0465926b473d91583761f2e63535adc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.3ex; height:2.843ex;" alt="{\displaystyle A_{j},}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1,\dots ,n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j=1,\dots ,n.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78fbe66f1d761a4c3189e16cfb5459335c8b20c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:12.465ex; height:2.509ex;" alt="{\displaystyle j=1,\dots ,n.}"> By construction of the <a href="/wiki/Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">integral</a>, the vector space of integrable simple functions is <a href="/wiki/Dense_set" title="Dense set">dense</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(S,\Sigma ,\mu ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(S,\Sigma ,\mu ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cddc3e08ec345cada944426025cf76ec45d44c41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.745ex; height:2.843ex;" alt="{\displaystyle L^{p}(S,\Sigma ,\mu ).}"> More can be said when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> is a <a href="/wiki/Normal_space" title="Normal space">normal</a> <a href="/wiki/Topological_space" title="Topological space">topological space</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }"> its <a href="/wiki/Borel_algebra" class="mw-redirect" title="Borel algebra">Borel 𝜎–algebra</a>. Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\subseteq S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>⊆</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\subseteq S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/813e4347f2e0bf2b35e5647030e7127eba7b0325" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.385ex; height:2.343ex;" alt="{\displaystyle V\subseteq S}"> is an open set with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (V)<\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (V)<\infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/776fd809f40abd63cbefda94e3863a72a4322e87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.067ex; height:2.843ex;" alt="{\displaystyle \mu (V)<\infty .}"> Then for every Borel set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in \Sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18496f0ae7c4589763102f30f1dd3bb02c0008c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.262ex; height:2.176ex;" alt="{\displaystyle A\in \Sigma }"> contained in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> there exist a closed set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> and an open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\subseteq A\subseteq U\subseteq V\quad {\text{and}}\quad \mu (U\setminus F)=\mu (U)-\mu (F)<\varepsilon ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>⊆</mo> <mi>A</mi> <mo>⊆</mo> <mi>U</mi> <mo>⊆</mo> <mi>V</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo class="MJX-variant">∖</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>ε</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\subseteq A\subseteq U\subseteq V\quad {\text{and}}\quad \mu (U\setminus F)=\mu (U)-\mu (F)<\varepsilon ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47203f300fd6302a096a52c85e5fe51caa525a04" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.384ex; height:2.843ex;" alt="{\displaystyle F\subseteq A\subseteq U\subseteq V\quad {\text{and}}\quad \mu (U\setminus F)=\mu (U)-\mu (F)<\varepsilon ,}"> for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04ec3670b50384a3ce48aca42e7cc5131a06b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}">. Subsequently, there exists a <a href="/wiki/Urysohn_function" class="mw-redirect" title="Urysohn function">Urysohn function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq \varphi \leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>φ</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq \varphi \leq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ed62c4a618fc33d728fc1e18f4fa70c6651a50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.042ex; height:2.676ex;" alt="{\displaystyle 0\leq \varphi \leq 1}"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> that is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\setminus U,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo class="MJX-variant">∖</mo> <mi>U</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\setminus U,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe790af080f2df7cac1b28310d1d83fdd70c417f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.123ex; height:2.843ex;" alt="{\displaystyle S\setminus U,}"> with <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{S}|\mathbf {1} _{A}-\varphi |\,\mathrm {d} \mu <\varepsilon \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>−</mo> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>μ</mi> <mo><</mo> <mi>ε</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{S}|\mathbf {1} _{A}-\varphi |\,\mathrm {d} \mu <\varepsilon \,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5cffec35706eeed9bd0b1fabd45e22d8283febd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.725ex; height:5.676ex;" alt="{\displaystyle \int _{S}|\mathbf {1} _{A}-\varphi |\,\mathrm {d} \mu <\varepsilon \,.}"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> can be covered by an increasing sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (V_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (V_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68ec21219192d30e3e77e73971cb29f08af52001" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.383ex; height:2.843ex;" alt="{\displaystyle (V_{n})}"> of open sets that have finite measure, then the space of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">–integrable continuous functions is dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(S,\Sigma ,\mu ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(S,\Sigma ,\mu ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cddc3e08ec345cada944426025cf76ec45d44c41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.745ex; height:2.843ex;" alt="{\displaystyle L^{p}(S,\Sigma ,\mu ).}"> More precisely, one can use bounded continuous functions that vanish outside one of the open sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13599fba5547371dbdad5c025205c6876e491516" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.221ex; height:2.509ex;" alt="{\displaystyle V_{n}.}"> This applies in particular when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\mathbb {R} ^{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\mathbb {R} ^{d}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b0bf04d6266ea81c5279a23971eacd19af797b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.368ex; height:2.676ex;" alt="{\displaystyle S=\mathbb {R} ^{d}}"> and when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> is the Lebesgue measure. For example, the space of continuous and compactly supported functions as well as the space of integrable <a href="/wiki/Step_function" title="Step function">step functions</a> are dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\mathbb {R} ^{d})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\mathbb {R} ^{d})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1edd20896725cf8163babcbdec3bea28b76ba7b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.221ex; height:3.176ex;" alt="{\displaystyle L^{p}(\mathbb {R} ^{d})}">. <div class="mw-heading mw-heading3"><h3 id="Closed_subspaces">Closed subspaces</h3>[<a href="/w/index.php?title=Lp_space&action=edit&section=18" title="Edit section: Closed subspaces">edit</a>]</div> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<p<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<p<\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/193a8d9032e5280409ac09830d95141319b20799" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.853ex; height:2.509ex;" alt="{\displaystyle 0<p<\infty }"> is any positive real number, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> is a <a href="/wiki/Probability_measure" title="Probability measure">probability measure</a> on a measurable space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S,\Sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S,\Sigma )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a345c3f30d6b3ee975e3afb6fa610cd3f67494e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.021ex; height:2.843ex;" alt="{\displaystyle (S,\Sigma )}"> (so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty }(\mu )\subseteq L^{p}(\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>⊆</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }(\mu )\subseteq L^{p}(\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95c3f6cd75c8bfad8c9f3f1de2fd2eb475908037" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.621ex; height:2.843ex;" alt="{\displaystyle L^{\infty }(\mu )\subseteq L^{p}(\mu )}">), and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\subseteq L^{\infty }(\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>⊆</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\subseteq L^{\infty }(\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7e60952918eab3c219306cd6ce75ffb9430282d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.555ex; height:2.843ex;" alt="{\displaystyle V\subseteq L^{\infty }(\mu )}"> is a vector subspace, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> is a closed subspace of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12496547ebfe44b8cb35e60d89069734ed4154c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.853ex; height:2.843ex;" alt="{\displaystyle L^{p}(\mu )}"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> is finite-dimensional<a href="#cite_note-FOOTNOTERudin1991117–119-21">[13]</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> was chosen independent of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">). In this theorem, which is due to <a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a>,<a href="#cite_note-FOOTNOTERudin1991117–119-21">[13]</a> it is crucial that the vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> be a subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ab400cc4dfd865180cd84c72dc894ca457671f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.458ex; height:2.343ex;" alt="{\displaystyle L^{\infty }}"> since it is possible to construct an infinite-dimensional closed vector subspace of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}\left(S^{1},{\tfrac {1}{2\pi }}\lambda \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mstyle> </mrow> <mi>λ</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{1}\left(S^{1},{\tfrac {1}{2\pi }}\lambda \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8b55d113374f2f8d2c2295e4803607f217fc7a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.365ex; height:4.843ex;" alt="{\displaystyle L^{1}\left(S^{1},{\tfrac {1}{2\pi }}\lambda \right)}"> (which is even a subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/291da6b7ddb8f116ad4b0975c8a717b6858c9ead" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{4}}">), where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"> is <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a> on the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60796c8d0c03cf575637d3202463b214d9635880" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{1}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2\pi }}\lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mstyle> </mrow> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2\pi }}\lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0161c512db8b6b98ce8d1631befdddc77cee1e5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.955ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2\pi }}\lambda }"> is the probability measure that results from dividing it by its mass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda (S^{1})=2\pi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo stretchy="false">(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda (S^{1})=2\pi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b28f7f8ef76ca3c4b312a4d4ae7913cf8682e782" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.98ex; height:3.176ex;" alt="{\displaystyle \lambda (S^{1})=2\pi .}"><a href="#cite_note-FOOTNOTERudin1991117–119-21">[13]</a> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Lp_space&action=edit&section=19" title="Edit section: Applications">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Statistics">Statistics</h3>[<a href="/w/index.php?title=Lp_space&action=edit&section=20" title="Edit section: Statistics">edit</a>]</div> In statistics, measures of <a href="/wiki/Central_tendency" title="Central tendency">central tendency</a> and <a href="/wiki/Statistical_dispersion" title="Statistical dispersion">statistical dispersion</a>, such as the <a href="/wiki/Mean" title="Mean">mean</a>, <a href="/wiki/Median" title="Median">median</a>, and <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a>, can be defined in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"> metrics, and measures of central tendency can be characterized as <a href="/wiki/Central_tendency#Solutions_to_variational_problems" title="Central tendency">solutions to variational problems</a>. In <a href="/wiki/Penalized_regression" class="mw-redirect" title="Penalized regression">penalized regression</a>, "L1 penalty" and "L2 penalty" refer to penalizing either the <a href="/wiki/Taxicab_geometry" title="Taxicab geometry"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74c288d1089f1ec85b01b4de25c441fc792bd2d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{1}}"> norm</a> of a solution's vector of parameter values (i.e. the sum of its absolute values), or its squared <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba162c66ca85776c83557af5088cc6f8584d1912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}"> norm (its <a href="/wiki/Euclidean_norm" class="mw-redirect" title="Euclidean norm">Euclidean length</a>). Techniques which use an L1 penalty, like <a href="/wiki/LASSO" class="mw-redirect" title="LASSO">LASSO</a>, encourage sparse solutions (where the many parameters are zero).<a href="#cite_note-22">[14]</a> <a href="/wiki/Elastic_net_regularization" title="Elastic net regularization">Elastic net regularization</a> uses a penalty term that is a combination of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74c288d1089f1ec85b01b4de25c441fc792bd2d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{1}}"> norm and the squared <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba162c66ca85776c83557af5088cc6f8584d1912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}"> norm of the parameter vector. <div class="mw-heading mw-heading3"><h3 id="Hausdorff–Young_inequality">Hausdorff–Young inequality</h3>[<a href="/w/index.php?title=Lp_space&action=edit&section=21" title="Edit section: Hausdorff–Young inequality">edit</a>]</div> The <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> for the real line (or, for <a href="/wiki/Periodic_functions" class="mw-redirect" title="Periodic functions">periodic functions</a>, see <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a>), maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81b25cc9016efea65c3a2be0b1a358b0d399ce3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.129ex; height:2.843ex;" alt="{\displaystyle L^{p}(\mathbb {R} )}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{q}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{q}(\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cfbbf0e75bd99b8e61b7acd2ffe9ce8bb97979a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.059ex; height:2.843ex;" alt="{\displaystyle L^{q}(\mathbb {R} )}"> (or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\mathbf {T} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\mathbf {T} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d99e7502b90f1ff0bdeee4f902b821dbaab10370" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.311ex; height:2.843ex;" alt="{\displaystyle L^{p}(\mathbf {T} )}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{q}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/174b9ba5de2319a7cca1be35d6262fb300355386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.958ex; height:2.343ex;" alt="{\displaystyle \ell ^{q}}">) respectively, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p\leq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq p\leq 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c94d35b201f4a50dd0574a25b95071e29386cde3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.691ex; height:2.509ex;" alt="{\displaystyle 1\leq p\leq 2}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>q</mi> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d094b49fac63e404b53136da33af9c7c37e67f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.069ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1.}"> This is a consequence of the <a href="/wiki/Riesz%E2%80%93Thorin_theorem#Hausdorff–Young_inequality" title="Riesz–Thorin theorem">Riesz–Thorin interpolation theorem</a>, and is made precise with the <a href="/wiki/Hausdorff%E2%80%93Young_inequality" title="Hausdorff–Young inequality">Hausdorff–Young inequality</a>. By contrast, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p>2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>></mo> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p>2,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18cf3a200042a925e1c235035a73f98c02dc79ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.167ex; height:2.509ex;" alt="{\displaystyle p>2,}"> the Fourier transform does not map into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{q}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{q}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21d539f8ff5c54abda3d4d308d68b5d4ca8140c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.218ex; height:2.343ex;" alt="{\displaystyle L^{q}.}"> <div class="mw-heading mw-heading3"><h3 id="Hilbert_spaces">Hilbert spaces</h3>[<a href="/w/index.php?title=Lp_space&action=edit&section=22" title="Edit section: Hilbert spaces">edit</a>]</div> <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a> are central to many applications, from <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> to <a href="/wiki/Stochastic_calculus" title="Stochastic calculus">stochastic calculus</a>. The spaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba162c66ca85776c83557af5088cc6f8584d1912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91f1f909abd70bd3d8fff0f7ae1ac23052387e18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{2}}"> are both Hilbert spaces. In fact, by choosing a Hilbert basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89862747e88ca143e979241a9a243b5ef66ddc67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.422ex; height:2.509ex;" alt="{\displaystyle E,}"> i.e., a maximal orthonormal subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba162c66ca85776c83557af5088cc6f8584d1912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}"> or any Hilbert space, one sees that every Hilbert space is isometrically isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2}(E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{2}(E)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49679837c74ed3be5f0dfc88e141fdf79664e504" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.609ex; height:3.176ex;" alt="{\displaystyle \ell ^{2}(E)}"> (same <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> as above), i.e., a Hilbert space of type <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11e5f61f7f39a2ef38b3466db6274c425e782404" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.671ex; height:2.676ex;" alt="{\displaystyle \ell ^{2}.}"> <div class="mw-heading mw-heading2"><h2 id="Generalizations_and_extensions">Generalizations and extensions</h2>[<a href="/w/index.php?title=Lp_space&action=edit&section=23" title="Edit section: Generalizations and extensions">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Weak_Lp">Weak Lp</h3>[<a href="/w/index.php?title=Lp_space&action=edit&section=24" title="Edit section: Weak Lp">edit</a>]</div> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S,\Sigma ,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S,\Sigma ,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cf1dfd7f18980638e424a78b4f5fe40df597c8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.456ex; height:2.843ex;" alt="{\displaystyle (S,\Sigma ,\mu )}"> be a measure space, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> a <a href="/wiki/Measurable_function" title="Measurable function">measurable function</a> with real or complex values on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23bbb1f0f6ebdfa78b4fed06049640f7386bb44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.146ex; height:2.176ex;" alt="{\displaystyle S.}"> The <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">distribution function</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is defined for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\geq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/248525429e9cd266f53ab8c52d17bc206c546060" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.101ex; height:2.343ex;" alt="{\displaystyle t\geq 0}"> by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{f}(t)=\mu \{x\in S:|f(x)|>t\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>μ</mi> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>S</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mi>t</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{f}(t)=\mu \{x\in S:|f(x)|>t\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a4328450b0d5589af9a0df24a13be1a822bcb58" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.868ex; height:3.009ex;" alt="{\displaystyle \lambda _{f}(t)=\mu \{x\in S:|f(x)|>t\}.}"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(S,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(S,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d5a4c54613c1c773a1caa81a731c340f546e320" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.386ex; height:2.843ex;" alt="{\displaystyle L^{p}(S,\mu )}"> for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p<\infty ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq p<\infty ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e72ecb9ef3faf51ea54b15aabd9fe02b82179cfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.499ex; height:2.509ex;" alt="{\displaystyle 1\leq p<\infty ,}"> then by <a href="/wiki/Markov%27s_inequality" title="Markov's inequality">Markov's inequality</a>, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{f}(t)\leq {\frac {\|f\|_{p}^{p}}{t^{p}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{f}(t)\leq {\frac {\|f\|_{p}^{p}}{t^{p}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c48fd53174874869a83e6f0adf03194bba1f0dc9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.738ex; height:5.843ex;" alt="{\displaystyle \lambda _{f}(t)\leq {\frac {\|f\|_{p}^{p}}{t^{p}}}}"> A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is said to be in the space weak <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(S,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(S,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d5a4c54613c1c773a1caa81a731c340f546e320" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.386ex; height:2.843ex;" alt="{\displaystyle L^{p}(S,\mu )}">, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p,w}(S,\mu ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>w</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p,w}(S,\mu ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2ed93d2abc725650612f07f7bf6c8e5a9335a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.667ex; height:2.843ex;" alt="{\displaystyle L^{p,w}(S,\mu ),}"> if there is a constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c84d4126c6df243734f9355927c026df6b0d3859" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.027ex; height:2.176ex;" alt="{\displaystyle C>0}"> such that, for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t>0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t>0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2159aad37e35b69930ad3b3afe15dab07ef52a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.747ex; height:2.509ex;" alt="{\displaystyle t>0,}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{f}(t)\leq {\frac {C^{p}}{t^{p}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{f}(t)\leq {\frac {C^{p}}{t^{p}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d995fc5fb346dcd3b4efb7280c4dfae04beabb56" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.932ex; height:5.343ex;" alt="{\displaystyle \lambda _{f}(t)\leq {\frac {C^{p}}{t^{p}}}}"> The best constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> for this inequality is the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p,w}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>w</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p,w}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cada5717fe7c4d33e29cc43884c1b69ce32cf74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.276ex; height:2.343ex;" alt="{\displaystyle L^{p,w}}">-norm of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e9687ea22c0f310582e97ee5f6c6a5fca28203d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.925ex; height:2.509ex;" alt="{\displaystyle f,}"> and is denoted by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{p,w}=\sup _{t>0}~t\lambda _{f}^{1/p}(t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>w</mi> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>></mo> <mn>0</mn> </mrow> </munder> <mtext> </mtext> <mi>t</mi> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{p,w}=\sup _{t>0}~t\lambda _{f}^{1/p}(t).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac8a4b6f768603fb59450c7886b587d7f598ae1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.058ex; height:5.176ex;" alt="{\displaystyle \|f\|_{p,w}=\sup _{t>0}~t\lambda _{f}^{1/p}(t).}"> The weak <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"> coincide with the <a href="/wiki/Lorentz_space" title="Lorentz space">Lorentz spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p,\infty },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p,\infty },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768f3de70b4b9f0371d2b5d731c711c5c3dc79e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.389ex; height:2.676ex;" alt="{\displaystyle L^{p,\infty },}"> so this notation is also used to denote them. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p,w}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>w</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p,w}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cada5717fe7c4d33e29cc43884c1b69ce32cf74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.276ex; height:2.343ex;" alt="{\displaystyle L^{p,w}}">-norm is not a true norm, since the <a href="/wiki/Triangle_inequality" title="Triangle inequality">triangle inequality</a> fails to hold. Nevertheless, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(S,\mu ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(S,\mu ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e33715d4c8c783d527801f8bbbf6a80ffad62204" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.033ex; height:2.843ex;" alt="{\displaystyle L^{p}(S,\mu ),}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{p,w}\leq \|f\|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>w</mi> </mrow> </msub> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{p,w}\leq \|f\|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a76bc656266d3ffa20814e3b95ddf75d028f831c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.058ex; height:3.009ex;" alt="{\displaystyle \|f\|_{p,w}\leq \|f\|_{p}}"> and in particular <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(S,\mu )\subset L^{p,w}(S,\mu ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>⊂</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>w</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(S,\mu )\subset L^{p,w}(S,\mu ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/960d0ad8f3d89e2f53b195f92ed3a2844867e4cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.152ex; height:2.843ex;" alt="{\displaystyle L^{p}(S,\mu )\subset L^{p,w}(S,\mu ).}"> In fact, one has <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{L^{p}}^{p}=\int |f(x)|^{p}d\mu (x)\geq \int _{\{|f(x)|>t\}}t^{p}+\int _{\{|f(x)|\leq t\}}|f|^{p}\geq t^{p}\mu (\{|f|>t\}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mi>d</mi> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mi>t</mi> <mo fence="false" stretchy="false">}</mo> </mrow> </msub> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>t</mi> <mo fence="false" stretchy="false">}</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>≥</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mi>μ</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mi>t</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{L^{p}}^{p}=\int |f(x)|^{p}d\mu (x)\geq \int _{\{|f(x)|>t\}}t^{p}+\int _{\{|f(x)|\leq t\}}|f|^{p}\geq t^{p}\mu (\{|f|>t\}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1be7432e448010448df7e65528dcbe0697f60b48" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:72.089ex; height:6.009ex;" alt="{\displaystyle \|f\|_{L^{p}}^{p}=\int |f(x)|^{p}d\mu (x)\geq \int _{\{|f(x)|>t\}}t^{p}+\int _{\{|f(x)|\leq t\}}|f|^{p}\geq t^{p}\mu (\{|f|>t\}),}"> and raising to power <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cdfd6eb8d2c6f424b698d06aa99d31895c47e91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.494ex; height:2.843ex;" alt="{\displaystyle 1/p}"> and taking the supremum in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> one has <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{L^{p}}\geq \sup _{t>0}t\;\mu (\{|f|>t\})^{1/p}=\|f\|_{L^{p,w}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mrow> </msub> <mo>≥</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>></mo> <mn>0</mn> </mrow> </munder> <mi>t</mi> <mspace width="thickmathspace" /> <mi>μ</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mi>t</mi> <mo fence="false" stretchy="false">}</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>w</mi> </mrow> </msup> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{L^{p}}\geq \sup _{t>0}t\;\mu (\{|f|>t\})^{1/p}=\|f\|_{L^{p,w}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0365e6788ee068614932155ee3ea05757c2a4e4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.874ex; height:5.009ex;" alt="{\displaystyle \|f\|_{L^{p}}\geq \sup _{t>0}t\;\mu (\{|f|>t\})^{1/p}=\|f\|_{L^{p,w}}.}"> Under the convention that two functions are equal if they are equal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> almost everywhere, then the spaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p,w}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>w</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p,w}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cada5717fe7c4d33e29cc43884c1b69ce32cf74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.276ex; height:2.343ex;" alt="{\displaystyle L^{p,w}}"> are complete (<a href="#CITEREFGrafakos2004">Grafakos 2004</a>). For any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<r<p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>r</mi> <mo><</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<r<p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b45e60f780a1e406fd2ceb05d0e72eb1e71b74b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.577ex; height:2.509ex;" alt="{\displaystyle 0<r<p}"> the expression <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \||f|\|_{L^{p,\infty }}=\sup _{0<\mu (E)<\infty }\mu (E)^{-1/r+1/p}\left(\int _{E}|f|^{r}\,d\mu \right)^{1/r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo><</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>E</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>μ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \||f|\|_{L^{p,\infty }}=\sup _{0<\mu (E)<\infty }\mu (E)^{-1/r+1/p}\left(\int _{E}|f|^{r}\,d\mu \right)^{1/r}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57561ba795b6916ab5b7f3474ddaa6eb6f8b57fc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:49.213ex; height:7.176ex;" alt="{\displaystyle \||f|\|_{L^{p,\infty }}=\sup _{0<\mu (E)<\infty }\mu (E)^{-1/r+1/p}\left(\int _{E}|f|^{r}\,d\mu \right)^{1/r}}"> is comparable to the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p,w}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>w</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p,w}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cada5717fe7c4d33e29cc43884c1b69ce32cf74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.276ex; height:2.343ex;" alt="{\displaystyle L^{p,w}}">-norm. Further in the case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p>1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>></mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p>1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b18ad57c656bab68c7bd45e2bf4596d09de7c0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.167ex; height:2.509ex;" alt="{\displaystyle p>1,}"> this expression defines a norm if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b58a38e59ab46b19e83056796cc12fc9952d9e48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.956ex; height:2.176ex;" alt="{\displaystyle r=1.}"> Hence for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f127e7a5f2449ddf3edb8164c2d2439120641f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p>1}"> the weak <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"> spaces are <a href="/wiki/Banach_space" title="Banach space">Banach spaces</a> (<a href="#CITEREFGrafakos2004">Grafakos 2004</a>). A major result that uses the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p,w}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>w</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p,w}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cada5717fe7c4d33e29cc43884c1b69ce32cf74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.276ex; height:2.343ex;" alt="{\displaystyle L^{p,w}}">-spaces is the <a href="/wiki/Marcinkiewicz_interpolation" class="mw-redirect" title="Marcinkiewicz interpolation">Marcinkiewicz interpolation theorem</a>, which has broad applications to <a href="/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a> and the study of <a href="/wiki/Singular_integrals" class="mw-redirect" title="Singular integrals">singular integrals</a>. <div class="mw-heading mw-heading3"><h3 id="Weighted_Lp_spaces">Weighted Lp spaces</h3>[<a href="/w/index.php?title=Lp_space&action=edit&section=25" title="Edit section: Weighted Lp spaces">edit</a>]</div> As before, consider a <a href="/wiki/Measure_space" title="Measure space">measure space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S,\Sigma ,\mu ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S,\Sigma ,\mu ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/718f1ce681c5e8a48858708a31f1cac5e8c7cc99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.103ex; height:2.843ex;" alt="{\displaystyle (S,\Sigma ,\mu ).}"> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w:S\to [a,\infty ),a>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo>:</mo> <mi>S</mi> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>a</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w:S\to [a,\infty ),a>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be42e0dfec9b030fa43d1c342a1d1e5be831f1fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.378ex; height:2.843ex;" alt="{\displaystyle w:S\to [a,\infty ),a>0}"> be a measurable function. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}">-weighted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"> space is defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(S,w\,\mathrm {d} \mu ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>w</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(S,w\,\mathrm {d} \mu ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c98f48ff9665fa8a9c07307c8ea92bcaffc12cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.377ex; height:2.843ex;" alt="{\displaystyle L^{p}(S,w\,\mathrm {d} \mu ),}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w\,\mathrm {d} \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w\,\mathrm {d} \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26fe063f9419d89a796f86c848b0976b981b46da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.676ex;" alt="{\displaystyle w\,\mathrm {d} \mu }"> means the measure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"> defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (A)\equiv \int _{A}w(x)\,\mathrm {d} \mu (x),\qquad A\in \Sigma ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mi>w</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="2em" /> <mi>A</mi> <mo>∈</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu (A)\equiv \int _{A}w(x)\,\mathrm {d} \mu (x),\qquad A\in \Sigma ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76e49861367adf55565afb3d7946fec10060bd86" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.639ex; height:5.676ex;" alt="{\displaystyle \nu (A)\equiv \int _{A}w(x)\,\mathrm {d} \mu (x),\qquad A\in \Sigma ,}"> or, in terms of the <a href="/wiki/Radon%E2%80%93Nikodym_theorem" title="Radon–Nikodym theorem">Radon–Nikodym derivative</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w={\tfrac {\mathrm {d} \nu }{\mathrm {d} \mu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ν</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>μ</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w={\tfrac {\mathrm {d} \nu }{\mathrm {d} \mu }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5edf01420801b50413d3a6c6ddabcbcb392d723" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:7.504ex; height:4.176ex;" alt="{\displaystyle w={\tfrac {\mathrm {d} \nu }{\mathrm {d} \mu }}}"> the <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(S,w\,\mathrm {d} \mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>w</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(S,w\,\mathrm {d} \mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f53de3fac5567030b8ef9e3d1d4ee4042f959435" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.73ex; height:2.843ex;" alt="{\displaystyle L^{p}(S,w\,\mathrm {d} \mu )}"> is explicitly <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|u\|_{L^{p}(S,w\,\mathrm {d} \mu )}\equiv \left(\int _{S}w(x)|u(x)|^{p}\,\mathrm {d} \mu (x)\right)^{1/p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>u</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>w</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>≡</mo> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mi>w</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|u\|_{L^{p}(S,w\,\mathrm {d} \mu )}\equiv \left(\int _{S}w(x)|u(x)|^{p}\,\mathrm {d} \mu (x)\right)^{1/p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8359475653cdb052fb1f9ce0400de3e59f14b747" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.147ex; height:6.676ex;" alt="{\displaystyle \|u\|_{L^{p}(S,w\,\mathrm {d} \mu )}\equiv \left(\int _{S}w(x)|u(x)|^{p}\,\mathrm {d} \mu (x)\right)^{1/p}}"> As <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}">-spaces, the weighted spaces have nothing special, since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(S,w\,\mathrm {d} \mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>w</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(S,w\,\mathrm {d} \mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f53de3fac5567030b8ef9e3d1d4ee4042f959435" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.73ex; height:2.843ex;" alt="{\displaystyle L^{p}(S,w\,\mathrm {d} \mu )}"> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(S,\mathrm {d} \nu ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ν</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(S,\mathrm {d} \nu ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b5e007e687074d037ef461768bd71f0fda36034" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.156ex; height:2.843ex;" alt="{\displaystyle L^{p}(S,\mathrm {d} \nu ).}"> But they are the natural framework for several results in harmonic analysis (<a href="#CITEREFGrafakos2004">Grafakos 2004</a>); they appear for example in the <a href="/wiki/Muckenhoupt_weights" title="Muckenhoupt weights">Muckenhoupt theorem</a>: for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1<p<\infty ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1<p<\infty ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/203e39618106a0a92baad7c6443a390d1dec4792" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.499ex; height:2.509ex;" alt="{\displaystyle 1<p<\infty ,}"> the classical <a href="/wiki/Hilbert_transform" title="Hilbert transform">Hilbert transform</a> is defined on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\mathbf {T} ,\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\mathbf {T} ,\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df08e809c1f050b851e34f5f8950e1518cfa326a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.7ex; height:2.843ex;" alt="{\displaystyle L^{p}(\mathbf {T} ,\lambda )}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {T} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9593e3b995a1b57c078873a5ea186c7012e1a5ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.859ex; height:2.176ex;" alt="{\displaystyle \mathbf {T} }"> denotes the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"> the Lebesgue measure; the (nonlinear) <a href="/wiki/Hardy%E2%80%93Littlewood_maximal_operator" class="mw-redirect" title="Hardy–Littlewood maximal operator">Hardy–Littlewood maximal operator</a> is bounded on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\mathbb {R} ^{n},\lambda ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\mathbb {R} ^{n},\lambda ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398becaf4dcc6cf545857dda3017cd286feecb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.384ex; height:2.843ex;" alt="{\displaystyle L^{p}(\mathbb {R} ^{n},\lambda ).}"> Muckenhoupt's theorem describes weights <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}"> such that the Hilbert transform remains bounded on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\mathbf {T} ,w\,\mathrm {d} \lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>,</mo> <mi>w</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\mathbf {T} ,w\,\mathrm {d} \lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dcbb15b8926e8a2b55f3088ae400aacba37f703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.043ex; height:2.843ex;" alt="{\displaystyle L^{p}(\mathbf {T} ,w\,\mathrm {d} \lambda )}"> and the maximal operator on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\mathbb {R} ^{n},w\,\mathrm {d} \lambda ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <mi>w</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\mathbb {R} ^{n},w\,\mathrm {d} \lambda ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc0c0f644d9aeaa1b820932984de8ce150202a3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.728ex; height:2.843ex;" alt="{\displaystyle L^{p}(\mathbb {R} ^{n},w\,\mathrm {d} \lambda ).}"> <div class="mw-heading mw-heading3"><h3 id="Lp_spaces_on_manifolds">Lp spaces on manifolds</h3>[<a href="/w/index.php?title=Lp_space&action=edit&section=26" title="Edit section: Lp spaces on manifolds">edit</a>]</div> One may also define spaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(M)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7106da28df5714bb538e6f840c8fd8d6f0821afc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.894ex; height:2.843ex;" alt="{\displaystyle L^{p}(M)}"> on a manifold, called the intrinsic <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"> spaces of the manifold, using <a href="/wiki/Density_on_a_manifold" title="Density on a manifold">densities</a>. <div class="mw-heading mw-heading3"><h3 id="Vector-valued_Lp_spaces">Vector-valued Lp spaces</h3>[<a href="/w/index.php?title=Lp_space&action=edit&section=27" title="Edit section: Vector-valued Lp spaces">edit</a>]</div> Given a measure space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,\Sigma ,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega ,\Sigma ,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22ed3b570455214147b8afded02ac578a0ba86e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.635ex; height:2.843ex;" alt="{\displaystyle (\Omega ,\Sigma ,\mu )}"> and a <a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">locally convex space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> (here assumed to be <a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">complete</a>), it is possible to define spaces of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-integrable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}">-valued functions on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"> in a number of ways. One way is to define the spaces of <a href="/wiki/Bochner_integral" title="Bochner integral">Bochner integrable</a> and <a href="/wiki/Pettis_integral" title="Pettis integral">Pettis integrable</a> functions, and then endow them with <a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">locally convex</a> <a href="/wiki/Vector_topology" class="mw-redirect" title="Vector topology">TVS-topologies</a> that are (each in their own way) a natural generalization of the usual <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"> topology. Another way involves <a href="/wiki/Topological_tensor_product" title="Topological tensor product">topological tensor products</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\Omega ,\Sigma ,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\Omega ,\Sigma ,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/452ff5713724a16623e37f0eabc77b4e3b1fd768" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.277ex; height:2.843ex;" alt="{\displaystyle L^{p}(\Omega ,\Sigma ,\mu )}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a2566d01f104ef084ea424b8b35c2534f7f902b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.422ex; height:2.176ex;" alt="{\displaystyle E.}"> Element of the vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\Omega ,\Sigma ,\mu )\otimes E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>⊗</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\Omega ,\Sigma ,\mu )\otimes E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5100a7ded4cbbf4ab51e574e53f1193d83f1e95e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.893ex; height:2.843ex;" alt="{\displaystyle L^{p}(\Omega ,\Sigma ,\mu )\otimes E}"> are finite sums of simple tensors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}\otimes e_{1}+\cdots +f_{n}\otimes e_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⊗</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}\otimes e_{1}+\cdots +f_{n}\otimes e_{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fa1ccc36b5cd567cdb2a3dc3840a35c80171870" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.723ex; height:2.509ex;" alt="{\displaystyle f_{1}\otimes e_{1}+\cdots +f_{n}\otimes e_{n},}"> where each simple tensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\times e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>×</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\times e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15d46c762a16d66984120b0a518e91f3dcec2bd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.202ex; height:2.509ex;" alt="{\displaystyle f\times e}"> may be identified with the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega \to E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">→</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega \to E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b869a1473e2f9a820a3d59c25cdd81d1395b6910" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.068ex; height:2.176ex;" alt="{\displaystyle \Omega \to E}"> that sends <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto ef(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>e</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto ef(x).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7aea6c161f30304389b3ea799ffb97b78312886" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.092ex; height:2.843ex;" alt="{\displaystyle x\mapsto ef(x).}"> This <a href="/wiki/Tensor_product" title="Tensor product">tensor product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\Omega ,\Sigma ,\mu )\otimes E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>⊗</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\Omega ,\Sigma ,\mu )\otimes E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5100a7ded4cbbf4ab51e574e53f1193d83f1e95e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.893ex; height:2.843ex;" alt="{\displaystyle L^{p}(\Omega ,\Sigma ,\mu )\otimes E}"> is then endowed with a locally convex topology that turns it into a <a href="/wiki/Topological_tensor_product" title="Topological tensor product">topological tensor product</a>, the most common of which are the <a href="/wiki/Projective_tensor_product" title="Projective tensor product">projective tensor product</a>, denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\Omega ,\Sigma ,\mu )\otimes _{\pi }E,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msub> <mi>E</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\Omega ,\Sigma ,\mu )\otimes _{\pi }E,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d74a495d7a1ae54e6d51fc28c9b0d63baff7c908" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.714ex; height:2.843ex;" alt="{\displaystyle L^{p}(\Omega ,\Sigma ,\mu )\otimes _{\pi }E,}"> and the <a href="/wiki/Injective_tensor_product" title="Injective tensor product">injective tensor product</a>, denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\Omega ,\Sigma ,\mu )\otimes _{\varepsilon }E.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mi>E</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\Omega ,\Sigma ,\mu )\otimes _{\varepsilon }E.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da349b3c06629de4a53132135e65cf1ea4bef7f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.538ex; height:2.843ex;" alt="{\displaystyle L^{p}(\Omega ,\Sigma ,\mu )\otimes _{\varepsilon }E.}"> In general, neither of these space are complete so their <a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">completions</a> are constructed, which are respectively denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\Omega ,\Sigma ,\mu ){\widehat {\otimes }}_{\pi }E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>⊗</mo> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msub> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\Omega ,\Sigma ,\mu ){\widehat {\otimes }}_{\pi }E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dca91b05c47e934c841fb9469b68d8b8c1c7ac6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.035ex; height:3.176ex;" alt="{\displaystyle L^{p}(\Omega ,\Sigma ,\mu ){\widehat {\otimes }}_{\pi }E}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\Omega ,\Sigma ,\mu ){\widehat {\otimes }}_{\varepsilon }E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>⊗</mo> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\Omega ,\Sigma ,\mu ){\widehat {\otimes }}_{\varepsilon }E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5896a131b591f09a3461e147d2007f8a532de640" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.859ex; height:3.176ex;" alt="{\displaystyle L^{p}(\Omega ,\Sigma ,\mu ){\widehat {\otimes }}_{\varepsilon }E}"> (this is analogous to how the space of scalar-valued <a href="/wiki/Simple_function" title="Simple function">simple functions</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9e6a34b6c6b77b92a65f692aad0f65b10f5bf26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \Omega ,}"> when seminormed by any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{p},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{p},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/084d3b2c0095344e5c4ec2ebeb8ca233f854dc43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.71ex; height:3.009ex;" alt="{\displaystyle \|\cdot \|_{p},}"> is not complete so a completion is constructed which, after being quotiented by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ker \|\cdot \|_{p},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ker</mi> <mo>⁡</mo> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ker \|\cdot \|_{p},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87a5f754b84fdba85bd4d7d005b99f0f1e045dd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.269ex; height:3.009ex;" alt="{\displaystyle \ker \|\cdot \|_{p},}"> is isometrically isomorphic to the Banach space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(\Omega ,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(\Omega ,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5075016f83493f7cded328bb0a03a64b91e1cfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.565ex; height:2.843ex;" alt="{\displaystyle L^{p}(\Omega ,\mu )}">). <a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a> showed that when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> is a <a href="/wiki/Nuclear_space" title="Nuclear space">nuclear space</a> (a concept he introduced), then these two constructions are, respectively, canonically TVS-isomorphic with the spaces of Bochner and Pettis integral functions mentioned earlier; in short, they are indistinguishable. <div class="mw-heading mw-heading3"><h3 id="L0_space_of_measurable_functions">L0 space of measurable functions</h3>[<a href="/w/index.php?title=Lp_space&action=edit&section=28" title="Edit section: L0 space of measurable functions">edit</a>]</div> The vector space of (<a href="/wiki/Equivalence_class" title="Equivalence class">equivalence classes</a> of) measurable functions on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S,\Sigma ,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S,\Sigma ,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cf1dfd7f18980638e424a78b4f5fe40df597c8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.456ex; height:2.843ex;" alt="{\displaystyle (S,\Sigma ,\mu )}"> is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{0}(S,\Sigma ,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{0}(S,\Sigma ,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/094893016c59257a7e50e52e8306f070b0330450" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.093ex; height:3.176ex;" alt="{\displaystyle L^{0}(S,\Sigma ,\mu )}"> (<a href="#CITEREFKaltonPeckRoberts1984">Kalton, Peck & Roberts 1984</a>). By definition, it contains all the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/842a68b4ad5a08d1837cb27a4fbbea432e045716" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.289ex; height:2.676ex;" alt="{\displaystyle L^{p},}"> and is equipped with the topology of <a href="/wiki/Convergence_in_measure" title="Convergence in measure">convergence in measure</a>. When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> is a probability measure (i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (S)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (S)=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53bb43e64a447d421fe01deed4921e6b47f1c19a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.971ex; height:2.843ex;" alt="{\displaystyle \mu (S)=1}">), this mode of convergence is named <a href="/wiki/Convergence_in_probability" class="mw-redirect" title="Convergence in probability">convergence in probability</a>. The space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50d9c756a343edb4557be976651d65368e888b55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{0}}"> is always a <a href="/wiki/Topological_abelian_group" title="Topological abelian group">topological abelian group</a> but is only a <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector space</a> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (S)<\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (S)<\infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7c764db37d07d2527c69f56b615ae63315e585d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.779ex; height:2.843ex;" alt="{\displaystyle \mu (S)<\infty .}"> This is because scalar multiplication is continuous if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (S)<\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (S)<\infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7c764db37d07d2527c69f56b615ae63315e585d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.779ex; height:2.843ex;" alt="{\displaystyle \mu (S)<\infty .}"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S,\Sigma ,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S,\Sigma ,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cf1dfd7f18980638e424a78b4f5fe40df597c8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.456ex; height:2.843ex;" alt="{\displaystyle (S,\Sigma ,\mu )}"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }">-finite then the <a href="/wiki/Weaker_topology" class="mw-redirect" title="Weaker topology">weaker topology</a> of <a href="/wiki/Local_convergence_in_measure" class="mw-redirect" title="Local convergence in measure">local convergence in measure</a> is an <a href="/wiki/F-space" title="F-space">F-space</a>, i.e. a <a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">completely</a> <a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">metrizable topological vector space</a>. Moreover, this topology is isometric to global convergence in measure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S,\Sigma ,\nu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S,\Sigma ,\nu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63d72c8201a667270cc6b56071e9c3edaf83762f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.287ex; height:2.843ex;" alt="{\displaystyle (S,\Sigma ,\nu )}"> for a suitable choice of <a href="/wiki/Probability_measure" title="Probability measure">probability measure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12bafc5c30bf9727eb4c005ff0c0632d555b3b7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.879ex; height:1.676ex;" alt="{\displaystyle \nu .}"> The description is easier when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> is finite. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> is a <a href="/wiki/Finite_measure" title="Finite measure">finite measure</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S,\Sigma ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S,\Sigma ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85d93c18e5e18d3efa7b52bd091a1660fbcdb605" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.667ex; height:2.843ex;" alt="{\displaystyle (S,\Sigma ),}"> the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> function admits for the convergence in measure the following <a href="/wiki/Fundamental_system_of_neighborhoods" class="mw-redirect" title="Fundamental system of neighborhoods">fundamental system of neighborhoods</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\varepsilon }={\Bigl \{}f:\mu {\bigl (}\{x:|f(x)|>\varepsilon \}{\bigr )}<\varepsilon {\Bigr \}},\qquad \varepsilon >0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">{</mo> </mrow> </mrow> <mi>f</mi> <mo>:</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mi>ε</mi> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo><</mo> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">}</mo> </mrow> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>ε</mi> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\varepsilon }={\Bigl \{}f:\mu {\bigl (}\{x:|f(x)|>\varepsilon \}{\bigr )}<\varepsilon {\Bigr \}},\qquad \varepsilon >0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa48bf929c7b8b9786a4dd16013ce5c456bc5a56" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:46.637ex; height:4.843ex;" alt="{\displaystyle V_{\varepsilon }={\Bigl \{}f:\mu {\bigl (}\{x:|f(x)|>\varepsilon \}{\bigr )}<\varepsilon {\Bigr \}},\qquad \varepsilon >0.}"> The topology can be defined by any metric <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"> of the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(f,g)=\int _{S}\varphi {\bigl (}|f(x)-g(x)|{\bigr )}\,\mathrm {d} \mu (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(f,g)=\int _{S}\varphi {\bigl (}|f(x)-g(x)|{\bigr )}\,\mathrm {d} \mu (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7c96fe7e12cbc1a9080e08017eaf82ca5f48f7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.201ex; height:5.676ex;" alt="{\displaystyle d(f,g)=\int _{S}\varphi {\bigl (}|f(x)-g(x)|{\bigr )}\,\mathrm {d} \mu (x)}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> is bounded continuous concave and non-decreasing on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,\infty ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,\infty ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/149e364417799ba3a8c23e972f2e59d97fb1a6a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.718ex; height:2.843ex;" alt="{\displaystyle [0,\infty ),}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (0)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (0)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dd0eadcddc166a18e283a4c5ed3a51c1e923d3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.753ex; height:2.843ex;" alt="{\displaystyle \varphi (0)=0}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (t)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (t)>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39cb6adfc121c359a90e994cfd968a9b0b9eb883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.43ex; height:2.843ex;" alt="{\displaystyle \varphi (t)>0}"> when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29a2960e88369263fe3cfe00ccbfeb83daee212a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t>0}"> (for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (t)=\min(t,1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (t)=\min(t,1).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/431f4dc751168287ff2248002eb8d074e1eb6c0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.635ex; height:2.843ex;" alt="{\displaystyle \varphi (t)=\min(t,1).}"> Such a metric is called <a href="/wiki/Paul_L%C3%A9vy_(mathematician)" title="Paul Lévy (mathematician)">Lévy</a>-metric for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{0}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf584b475725f5928b3abf43885bea7fa7240f9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.284ex; height:2.676ex;" alt="{\displaystyle L^{0}.}"> Under this metric the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50d9c756a343edb4557be976651d65368e888b55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{0}}"> is complete. However, as mentioned above, scalar multiplication is continuous with respect to this metric only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (S)<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (S)<\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d64fc832b992b733a2211d77e3d1ea6f4894718" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.132ex; height:2.843ex;" alt="{\displaystyle \mu (S)<\infty }">. To see this, consider the Lebesgue measurable function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85e6e186aabef9e51814bbce62e625dc67e825f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} }"> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f690285952308aa49e3c6aac892df31cad6d1b06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.846ex; height:2.843ex;" alt="{\displaystyle f(x)=x}">. Then clearly <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{c\rightarrow 0}d(cf,0)=\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mi>d</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mi>f</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{c\rightarrow 0}d(cf,0)=\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b58eb370221ef3294a5f3ac92f4252193ed625ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.546ex; height:4.009ex;" alt="{\displaystyle \lim _{c\rightarrow 0}d(cf,0)=\infty }">. The space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50d9c756a343edb4557be976651d65368e888b55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{0}}"> is in general not locally bounded, and not locally convex. For the infinite Lebesgue measure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7035fcb9fe3ebecc6bc9f372f82d0352202c8bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{n},}"> the definition of the fundamental system of neighborhoods could be modified as follows <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{\varepsilon }=\left\{f:\lambda \left(\left\{x:|f(x)|>\varepsilon {\text{ and }}|x|<{\tfrac {1}{\varepsilon }}\right\}\right)<\varepsilon \right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mi>f</mi> <mo>:</mo> <mi>λ</mi> <mrow> <mo>(</mo> <mrow> <mo>{</mo> <mrow> <mi>x</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>ε</mi> </mfrac> </mstyle> </mrow> </mrow> <mo>}</mo> </mrow> <mo>)</mo> </mrow> <mo><</mo> <mi>ε</mi> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{\varepsilon }=\left\{f:\lambda \left(\left\{x:|f(x)|>\varepsilon {\text{ and }}|x|<{\tfrac {1}{\varepsilon }}\right\}\right)<\varepsilon \right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83e43fb7a7f26934b138a07e512fa11a9bde9ed7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:48.43ex; height:3.509ex;" alt="{\displaystyle W_{\varepsilon }=\left\{f:\lambda \left(\left\{x:|f(x)|>\varepsilon {\text{ and }}|x|<{\tfrac {1}{\varepsilon }}\right\}\right)<\varepsilon \right\}}"> The resulting space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{0}(\mathbb {R} ^{n},\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{0}(\mathbb {R} ^{n},\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e994da01ca0228c623863567d355002257ea3c2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.732ex; height:3.176ex;" alt="{\displaystyle L^{0}(\mathbb {R} ^{n},\lambda )}">, with the topology of local convergence in measure, is isomorphic to the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{0}(\mathbb {R} ^{n},g\,\lambda ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <mi>g</mi> <mspace width="thinmathspace" /> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{0}(\mathbb {R} ^{n},g\,\lambda ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/978fe4821cd4a3351e6c3485d052afcb772ac205" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.882ex; height:3.176ex;" alt="{\displaystyle L^{0}(\mathbb {R} ^{n},g\,\lambda ),}"> for any positive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }">–integrable density <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23a3f421f58ef3bc6f9ec70e883e1496ff871e9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.763ex; height:2.009ex;" alt="{\displaystyle g.}"> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Lp_space&action=edit&section=29" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 30em;"> <ul><li><a href="/wiki/Absolutely_integrable_function" title="Absolutely integrable function">Absolutely integrable function</a> – function whose absolute value has finite integralPages displaying wikidata descriptions as a fallback</li> <li><a href="/wiki/Bochner_space" title="Bochner space">Bochner space</a> – Type of topological space</li> <li><a href="/wiki/Orlicz_space" title="Orlicz space">Orlicz space</a> – Type of function space</li> <li><a href="/wiki/Hardy_space" title="Hardy space">Hardy space</a> – Concept within complex analysis</li> <li><a href="/wiki/Riesz%E2%80%93Thorin_theorem" title="Riesz–Thorin theorem">Riesz–Thorin theorem</a> – Theorem on operator interpolation</li> <li><a href="/wiki/H%C3%B6lder_mean" class="mw-redirect" title="Hölder mean">Hölder mean</a> – N-th root of the arithmetic mean of the given numbers raised to the power nPages displaying short descriptions of redirect targets</li> <li><a href="/wiki/H%C3%B6lder_space" class="mw-redirect" title="Hölder space">Hölder space</a> – Type of continuity of a complex-valued functionPages displaying short descriptions of redirect targets</li> <li><a href="/wiki/Root_mean_square" title="Root mean square">Root mean square</a> – Square root of the mean square</li> <li><a href="/wiki/Least_absolute_deviations" title="Least absolute deviations">Least absolute deviations</a> – Statistical optimality criterion</li> <li><a href="/wiki/Locally_integrable_function" title="Locally integrable function">Locally integrable function</a> – Function which is integratable on its domain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(L_{\text{loc}}^{1}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>loc</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(L_{\text{loc}}^{1}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449a41034cc06bc19b7e1981649f7b2b35a59020" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.954ex; height:3.176ex;" alt="{\displaystyle \left(L_{\text{loc}}^{1}\right)}"></li> <li><a href="/wiki/Pontryagin_duality#Haar_measure" title="Pontryagin duality"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(G)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f8c993aade676f12e12958b967facfad51f9e69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.278ex; height:2.843ex;" alt="{\displaystyle L^{p}(G)}"> spaces over a locally compact group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></a> – Duality for locally compact abelian groups</li> <li><a href="/wiki/Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a> – Periodicity computation method</li> <li><a href="/wiki/List_of_Banach_spaces" title="List of Banach spaces">List of Banach spaces</a></li> <li><a href="/wiki/Minkowski_distance" title="Minkowski distance">Minkowski distance</a> – Mathematical metric in normed vector space</li> <li><a href="/wiki/L-infinity" title="L-infinity">L-infinity</a> – Space of bounded sequences</li> <li><a href="/wiki/Lp_sum" title="Lp sum">Lp sum</a> – Measure in functional analysis</li> <li><a href="/w/index.php?title=Cm_sum&action=edit&redlink=1" class="new" title="Cm sum (page does not exist)">Cm space</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Lp_space&action=edit&section=30" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMaddox1988" class="citation cs2">Maddox, I. J. (1988), Elements of Functional Analysis (2nd ed.), Cambridge: CUP</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Functional+Analysis&rft.place=Cambridge&rft.edition=2nd&rft.pub=CUP&rft.date=1988&rft.aulast=Maddox&rft.aufirst=I.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALp+space" class="Z3988">, page 16 </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> Rafael Dahmen, Gábor Lukács: Long colimits of topological groups I: Continuous maps and homeomorphisms. in: Topology and its Applications Nr. 270, 2020. Example 2.14 </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGarling2007" class="citation book cs1">Garling, D. J. H. (2007). Inequalities: A Journey into Linear Analysis. Cambridge University Press. p. 54. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-87624-7" title="Special:BookSources/978-0-521-87624-7"><bdi>978-0-521-87624-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Inequalities%3A+A+Journey+into+Linear+Analysis&rft.pages=54&rft.pub=Cambridge+University+Press&rft.date=2007&rft.isbn=978-0-521-87624-7&rft.aulast=Garling&rft.aufirst=D.+J.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALp+space" class="Z3988"> </li> <li id="cite_note-FOOTNOTERudin198765-7"><a href="#cite_ref-FOOTNOTERudin198765_7-0">^</a> <a href="#CITEREFRudin1987">Rudin 1987</a>, p. 65. </li> <li id="cite_note-FOOTNOTESteinShakarchi20122-12"><a href="#cite_ref-FOOTNOTESteinShakarchi20122_12-0">^</a> <a href="#CITEREFSteinShakarchi2012">Stein & Shakarchi 2012</a>, p. 2. </li> <li id="cite_note-FOOTNOTERudin199137-13"><a href="#cite_ref-FOOTNOTERudin199137_13-0">^</a> <a href="#CITEREFRudin1991">Rudin 1991</a>, p. 37. </li> <li id="cite_note-FOOTNOTEBahouriCheminDanchin20111–4-14"><a href="#cite_ref-FOOTNOTEBahouriCheminDanchin20111–4_14-0">^</a> <a href="#CITEREFBahouriCheminDanchin2011">Bahouri, Chemin & Danchin 2011</a>, pp. 1–4. </li> <li id="cite_note-FOOTNOTEBahouriCheminDanchin20114-15"><a href="#cite_ref-FOOTNOTEBahouriCheminDanchin20114_15-0">^</a> <a href="#CITEREFBahouriCheminDanchin2011">Bahouri, Chemin & Danchin 2011</a>, p. 4. </li> <li id="cite_note-FOOTNOTEBahouriCheminDanchin20117–8-16">^ <a href="#cite_ref-FOOTNOTEBahouriCheminDanchin20117–8_16-0">a</a> <a href="#cite_ref-FOOTNOTEBahouriCheminDanchin20117–8_16-1">b</a> <a href="#cite_ref-FOOTNOTEBahouriCheminDanchin20117–8_16-2">c</a> <a href="#cite_ref-FOOTNOTEBahouriCheminDanchin20117–8_16-3">d</a> <a href="#cite_ref-FOOTNOTEBahouriCheminDanchin20117–8_16-4">e</a> <a href="#cite_ref-FOOTNOTEBahouriCheminDanchin20117–8_16-5">f</a> <a href="#CITEREFBahouriCheminDanchin2011">Bahouri, Chemin & Danchin 2011</a>, pp. 7–8. </li> <li id="cite_note-FOOTNOTERudin1987Theorem_6.16-18"><a href="#cite_ref-FOOTNOTERudin1987Theorem_6.16_18-0">^</a> <a href="#CITEREFRudin1987">Rudin 1987</a>, Theorem 6.16. </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchechter1997" class="citation cs2">Schechter, Eric (1997), Handbook of Analysis and its Foundations, London: Academic Press Inc.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Analysis+and+its+Foundations&rft.place=London&rft.pub=Academic+Press+Inc.&rft.date=1997&rft.aulast=Schechter&rft.aufirst=Eric&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALp+space" class="Z3988"> See Sections 14.77 and 27.44–47 </li> <li id="cite_note-VillaniEmbeddings-20"><a href="#cite_ref-VillaniEmbeddings_20-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVillani1985" class="citation cs2">Villani, Alfonso (1985), "Another note on the inclusion Lp(μ) ⊂ Lq(μ)", Amer. Math. Monthly, 92 (7): 485–487, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2322503">10.2307/2322503</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2322503">2322503</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0801221">0801221</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Amer.+Math.+Monthly&rft.atitle=Another+note+on+the+inclusion+%3Cspan+class%3D%22texhtml+%22+%3EL%3Csup%3Ep%3C%2Fsup%3E%28%CE%BC%29+%E2%8A%82+L%3Csup%3Eq%3C%2Fsup%3E%28%CE%BC%29%3C%2Fspan%3E&rft.volume=92&rft.issue=7&rft.pages=%3Cspan+class%3D%22nowrap%22%3E485-%3C%2Fspan%3E487&rft.date=1985&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D801221%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2322503%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2322503&rft.aulast=Villani&rft.aufirst=Alfonso&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALp+space" class="Z3988"> </li> <li id="cite_note-FOOTNOTERudin1991117–119-21">^ <a href="#cite_ref-FOOTNOTERudin1991117–119_21-0">a</a> <a href="#cite_ref-FOOTNOTERudin1991117–119_21-1">b</a> <a href="#cite_ref-FOOTNOTERudin1991117–119_21-2">c</a> <a href="#CITEREFRudin1991">Rudin 1991</a>, pp. 117–119. </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHastieTibshiraniWainwright2015" class="citation book cs1"><a href="/wiki/Trevor_Hastie" title="Trevor Hastie">Hastie, T. J.</a>; <a href="/wiki/Robert_Tibshirani" title="Robert Tibshirani">Tibshirani, R.</a>; Wainwright, M. J. (2015). Statistical Learning with Sparsity: The Lasso and Generalizations. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4987-1216-3" title="Special:BookSources/978-1-4987-1216-3"><bdi>978-1-4987-1216-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+Learning+with+Sparsity%3A+The+Lasso+and+Generalizations&rft.pub=CRC+Press&rft.date=2015&rft.isbn=978-1-4987-1216-3&rft.aulast=Hastie&rft.aufirst=T.+J.&rft.au=Tibshirani%2C+R.&rft.au=Wainwright%2C+M.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALp+space" class="Z3988"> </li> </ol></div></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-3"><a href="#cite_ref-3">^</a> The condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup \operatorname {range} |x|<+\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">sup</mo> <mi>range</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup \operatorname {range} |x|<+\infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe5a2895ba415c2df3bb1c78b6a3696d6126bd2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.337ex; height:2.843ex;" alt="{\displaystyle \sup \operatorname {range} |x|<+\infty .}"> is not equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup \operatorname {range} |x|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">sup</mo> <mi>range</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup \operatorname {range} |x|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54b31e0faf1bb580747f6bba39daed26b1d2f84f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.46ex; height:2.843ex;" alt="{\displaystyle \sup \operatorname {range} |x|}"> being finite, unless <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\neq \varnothing .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>≠</mo> <mi class="MJX-variant">∅</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\neq \varnothing .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19de91312324073eff89ed5ee49bbea2a2e1b83c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.533ex; height:2.676ex;" alt="{\displaystyle X\neq \varnothing .}"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\varnothing }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7af8fc701c3fb8064e7e0340ddfe7e2201795e17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.887ex; height:2.176ex;" alt="{\displaystyle X=\varnothing }"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup \operatorname {range} |x|=-\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">sup</mo> <mi>range</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup \operatorname {range} |x|=-\infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05da5547b9ddc82ce982fd3986c98a6f5af3962b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.337ex; height:2.843ex;" alt="{\displaystyle \sup \operatorname {range} |x|=-\infty .}"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> The definitions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{p},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{p},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/084d3b2c0095344e5c4ec2ebeb8ca233f854dc43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.71ex; height:3.009ex;" alt="{\displaystyle \|\cdot \|_{p},}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}^{p}(S,\,\mu ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}^{p}(S,\,\mu ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90b75589f561331ce2409ffa5dc1e384536d44cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.441ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}^{p}(S,\,\mu ),}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}(S,\,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}(S,\,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0806c3e125a89e0704774b4db9944b9c4f01bfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.773ex; height:2.843ex;" alt="{\displaystyle L^{p}(S,\,\mu )}"> can be extended to all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<p\leq \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo>≤</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<p\leq \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/915ecfea83a5b7e3432407d4eebea5a38f9948ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.853ex; height:2.509ex;" alt="{\displaystyle 0<p\leq \infty }"> (rather than just <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p\leq \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq p\leq \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7d63bb8c8def80f4fb709fbb2aae6a11c9cd41d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.853ex; height:2.509ex;" alt="{\displaystyle 1\leq p\leq \infty }">), but it is only when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p\leq \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq p\leq \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7d63bb8c8def80f4fb709fbb2aae6a11c9cd41d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.853ex; height:2.509ex;" alt="{\displaystyle 1\leq p\leq \infty }"> that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44e951f6e15fdb1bdd7d223bbbe628955251a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.063ex; height:3.009ex;" alt="{\displaystyle \|\cdot \|_{p}}"> is guaranteed to be a norm (although <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44e951f6e15fdb1bdd7d223bbbe628955251a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.063ex; height:3.009ex;" alt="{\displaystyle \|\cdot \|_{p}}"> is a <a href="/wiki/Quasi-seminorm" class="mw-redirect" title="Quasi-seminorm">quasi-seminorm</a> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<p\leq \infty ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo>≤</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<p\leq \infty ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a50201387608cbbf7b797fc15b7370c1d4fed1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.499ex; height:2.509ex;" alt="{\displaystyle 0<p\leq \infty ,}">). </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (S)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (S)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcc847bbfe75008987e1cb581a591f5cd7b6ca7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.971ex; height:2.843ex;" alt="{\displaystyle \mu (S)=0}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {esssup} |f|=-\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>esssup</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {esssup} |f|=-\infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e302f4ec596912888372d56edc3e3de5aa6e673" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.203ex; height:2.843ex;" alt="{\displaystyle \operatorname {esssup} |f|=-\infty .}"> </li> <li id="cite_note-Non0Value0Example-9">^ <a href="#cite_ref-Non0Value0Example_9-0">a</a> <a href="#cite_ref-Non0Value0Example_9-1">b</a> For example, if a non-empty measurable set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\neq \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>≠</mo> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\neq \varnothing }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2229dadebb0d164bac6427da68b6e30cf22ee7ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.97ex; height:2.676ex;" alt="{\displaystyle N\neq \varnothing }"> of measure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (N)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (N)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b976c47b5d229e78e08df8c63c14919a9c774e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.536ex; height:2.843ex;" alt="{\displaystyle \mu (N)=0}"> exists then its <a href="/wiki/Indicator_function" title="Indicator function">indicator function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} _{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{N}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92405b45fa8bf31efbfe9d79014d37ab768fa21d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.028ex; height:2.509ex;" alt="{\displaystyle \mathbf {1} _{N}}"> satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\mathbf {1} _{N}\|_{p}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {1} _{N}\|_{p}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2abb336b5f9c79129140839740d05b6f17bb0dcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.673ex; height:3.009ex;" alt="{\displaystyle \|\mathbf {1} _{N}\|_{p}=0}"> although <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} _{N}\neq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>≠</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{N}\neq 0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8b0bfc8138cf974619ca6e9dbd59371cca2a83f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.936ex; height:2.676ex;" alt="{\displaystyle \mathbf {1} _{N}\neq 0.}"> </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> Explicitly, the vector space operations are defined by: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(f+g)(x)&=f(x)+g(x),\\(sf)(x)&=sf(x)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>f</mi> <mo>+</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>s</mi> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>s</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(f+g)(x)&=f(x)+g(x),\\(sf)(x)&=sf(x)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43a3d5554d6f6cde8da719231be0079a955ffa08" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.193ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}(f+g)(x)&=f(x)+g(x),\\(sf)(x)&=sf(x)\end{aligned}}}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\in {\mathcal {L}}^{p}(S,\,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g\in {\mathcal {L}}^{p}(S,\,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2aca2d5f9fb9c6c59e54b1f766b5dff12f4d958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.063ex; height:2.843ex;" alt="{\displaystyle f,g\in {\mathcal {L}}^{p}(S,\,\mu )}"> and all scalars <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7be57a083bd9ebe931fad214b191cfb20227ff9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.737ex; height:1.676ex;" alt="{\displaystyle s.}"> These operations make <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44f87cc85cf5190b48fe6c61c049b0199f68a4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.794ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}"> into a vector space because if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"> is any scalar and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\in {\mathcal {L}}^{p}(S,\,\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g\in {\mathcal {L}}^{p}(S,\,\mu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2aca2d5f9fb9c6c59e54b1f766b5dff12f4d958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.063ex; height:2.843ex;" alt="{\displaystyle f,g\in {\mathcal {L}}^{p}(S,\,\mu )}"> then both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle sf}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle sf}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9546352f6401d2c99f1759ad9f1a895149405757" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.369ex; height:2.509ex;" alt="{\displaystyle sf}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f+g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>+</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f+g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d94a24abd865f6f9fd67a7df7e531cae1c769b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.235ex; height:2.509ex;" alt="{\displaystyle f+g}"> also belong to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}^{p}(S,\,\mu ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}^{p}(S,\,\mu ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff012d0786854669d5a299221e8ccc37ff540fa2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.441ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}^{p}(S,\,\mu ).}"> </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> This <a href="/wiki/Infimum" class="mw-redirect" title="Infimum">infimum</a> is attained by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{n};}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b549e70187f8870006c84782918965777bca240" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.705ex; height:2.343ex;" alt="{\displaystyle t_{n};}"> that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (f>t_{n})<2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>></mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo><</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (f>t_{n})<2^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aa965327f2aced154b136b9ff9d6b56ea9f0daf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.126ex; height:2.843ex;" alt="{\displaystyle \mu (f>t_{n})<2^{n}}"> holds. </li> </ol></div></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-UpperBoundForNormOfSum-11"><a href="#cite_ref-UpperBoundForNormOfSum_11-0">^</a> When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p<\infty ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq p<\infty ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e72ecb9ef3faf51ea54b15aabd9fe02b82179cfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.499ex; height:2.509ex;" alt="{\displaystyle 1\leq p<\infty ,}"> the inequality <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f+g\|_{p}^{p}\leq 2^{p-1}\left(\|f\|_{p}^{p}+\|g\|_{p}^{p}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mo>+</mo> <mi>g</mi> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <mo>≤</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>g</mi> <msubsup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f+g\|_{p}^{p}\leq 2^{p-1}\left(\|f\|_{p}^{p}+\|g\|_{p}^{p}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dcf1a60edeb669265823f0427a2284b7f63e543" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.239ex; height:3.343ex;" alt="{\displaystyle \|f+g\|_{p}^{p}\leq 2^{p-1}\left(\|f\|_{p}^{p}+\|g\|_{p}^{p}\right)}"> can be deduced from the fact that the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:[0,\infty )\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F:[0,\infty )\to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b97072779b79fe34db941084b7e1df5c9f99d17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.042ex; height:2.843ex;" alt="{\displaystyle F:[0,\infty )\to \mathbb {R} }"> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(t)=t^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(t)=t^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b81f939f0ad66fe18295db54bdc7a9a839ab784" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.387ex; height:2.843ex;" alt="{\displaystyle F(t)=t^{p}}"> is <a href="/wiki/Convex_function" title="Convex function">convex</a>, which by definition means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(tx+(1-t)y)\leq tF(x)+(1-t)F(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>t</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(tx+(1-t)y)\leq tF(x)+(1-t)F(y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83a19cd75ef97a8577c6f3f4cdb13c9ff25cfad0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.383ex; height:2.843ex;" alt="{\displaystyle F(tx+(1-t)y)\leq tF(x)+(1-t)F(y)}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq t\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>t</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq t\leq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86496f1001838495964ccca2851a6f29ff0c36a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.361ex; height:2.343ex;" alt="{\displaystyle 0\leq t\leq 1}"> and all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea0abffd33a692ded22accc104515a032851dff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.519ex; height:2.009ex;" alt="{\displaystyle x,y}"> in the domain of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6b34655c2ef19b56c81af0e6d1f2f6df0d3ed33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:2.176ex;" alt="{\displaystyle F.}"> Substituting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f|,|g|,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f|,|g|,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3af06505037ab179855b5777306c77ec9c2db46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.663ex; height:2.843ex;" alt="{\displaystyle |f|,|g|,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edef8290613648790a8ac1a95c2fb7c3972aea2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}}"> in for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ad6024a5a4f15ab696de3d6e4866da50c8a315d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.166ex; height:2.009ex;" alt="{\displaystyle x,y,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> gives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\tfrac {1}{2}}|f|+{\tfrac {1}{2}}|g|\right)^{p}\leq {\tfrac {1}{2}}|f|^{p}+{\tfrac {1}{2}}|g|^{p},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>g</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\tfrac {1}{2}}|f|+{\tfrac {1}{2}}|g|\right)^{p}\leq {\tfrac {1}{2}}|f|^{p}+{\tfrac {1}{2}}|g|^{p},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6beef0c0373ce1b91edba0074cbbb2ddb2795e5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:31.33ex; height:3.676ex;" alt="{\displaystyle \left({\tfrac {1}{2}}|f|+{\tfrac {1}{2}}|g|\right)^{p}\leq {\tfrac {1}{2}}|f|^{p}+{\tfrac {1}{2}}|g|^{p},}"> which proves that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (|f|+|g|)^{p}\leq 2^{p-1}(|f|^{p}+|g|^{p}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>≤</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>g</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (|f|+|g|)^{p}\leq 2^{p-1}(|f|^{p}+|g|^{p}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9dcde0b50a99040153041aae870d330387bf62e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.508ex; height:3.176ex;" alt="{\displaystyle (|f|+|g|)^{p}\leq 2^{p-1}(|f|^{p}+|g|^{p}).}"> The triangle inequality <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f+g|\leq |f|+|g|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo>+</mo> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f+g|\leq |f|+|g|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e91dd16c6a132b5ad6d7470555fe4227ef43f33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.449ex; height:2.843ex;" alt="{\displaystyle |f+g|\leq |f|+|g|}"> now implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f+g|^{p}\leq 2^{p-1}(|f|^{p}+|g|^{p}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo>+</mo> <mi>g</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>≤</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>g</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f+g|^{p}\leq 2^{p-1}(|f|^{p}+|g|^{p}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09381447aedbbd09c375faf0b939561316b6abb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.405ex; height:3.176ex;" alt="{\displaystyle |f+g|^{p}\leq 2^{p-1}(|f|^{p}+|g|^{p}).}"> The desired inequality follows by integrating both sides. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \blacksquare }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◼</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \blacksquare }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8733090f2d787d03101c3e16dc3f6404f0e7dd4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \blacksquare }"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Lp_space&action=edit&section=31" title="Edit section: References">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAdamsFournier2003" class="citation cs2">Adams, Robert A.; 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(2003), Sobolev Spaces (Second ed.), Academic Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-044143-3" title="Special:BookSources/978-0-12-044143-3"><bdi>978-0-12-044143-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sobolev+Spaces&rft.edition=Second&rft.pub=Academic+Press&rft.date=2003&rft.isbn=978-0-12-044143-3&rft.aulast=Adams&rft.aufirst=Robert+A.&rft.au=Fournier%2C+John+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALp+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBahouriCheminDanchin2011" class="citation book cs1"><a href="/wiki/Hajer_Bahouri" title="Hajer Bahouri">Bahouri, Hajer</a>; <a href="/wiki/Jean-Yves_Chemin" title="Jean-Yves Chemin">Chemin, Jean-Yves</a>; <a href="/w/index.php?title=Rapha%C3%ABl_Danchin&action=edit&redlink=1" class="new" title="Raphaël Danchin (page does not exist)">Danchin, Raphaël</a> (2011). <a rel="nofollow" class="external text" href="https://www.google.com/books?id=CcTnaveQkn0C">Fourier Analysis and Nonlinear Partial Differential Equations</a>. Grundlehren der mathematischen Wissenschaften. Vol. 343. Berlin, Heidelberg: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-16830-7" title="Special:BookSources/978-3-642-16830-7"><bdi>978-3-642-16830-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/704397128">704397128</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fourier+Analysis+and+Nonlinear+Partial+Differential+Equations&rft.place=Berlin%2C+Heidelberg&rft.series=Grundlehren+der+mathematischen+Wissenschaften&rft.pub=Springer&rft.date=2011&rft_id=info%3Aoclcnum%2F704397128&rft.isbn=978-3-642-16830-7&rft.aulast=Bahouri&rft.aufirst=Hajer&rft.au=Chemin%2C+Jean-Yves&rft.au=Danchin%2C+Rapha%C3%ABl&rft_id=https%3A%2F%2Fwww.google.com%2Fbooks%3Fid%3DCcTnaveQkn0C&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALp+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBourbaki1987" class="citation cs2"><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki, Nicolas</a> (1987), Topological vector spaces, Elements of mathematics, Berlin: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-13627-9" title="Special:BookSources/978-3-540-13627-9"><bdi>978-3-540-13627-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+vector+spaces&rft.place=Berlin&rft.series=Elements+of+mathematics&rft.pub=Springer-Verlag&rft.date=1987&rft.isbn=978-3-540-13627-9&rft.aulast=Bourbaki&rft.aufirst=Nicolas&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALp+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDiBenedetto2002" class="citation cs2">DiBenedetto, Emmanuele (2002), Real analysis, Birkhäuser, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-7643-4231-5" title="Special:BookSources/3-7643-4231-5"><bdi>3-7643-4231-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+analysis&rft.pub=Birkh%C3%A4user&rft.date=2002&rft.isbn=3-7643-4231-5&rft.aulast=DiBenedetto&rft.aufirst=Emmanuele&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALp+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDunfordSchwartz1958" class="citation cs2">Dunford, Nelson; Schwartz, Jacob T. (1958), Linear operators, volume I, Wiley-Interscience</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+operators%2C+volume+I&rft.pub=Wiley-Interscience&rft.date=1958&rft.aulast=Dunford&rft.aufirst=Nelson&rft.au=Schwartz%2C+Jacob+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALp+space" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDuren1970" class="citation cs2">Duren, P. 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New York, NY: <a href="/wiki/McGraw-Hill_Science/Engineering/Math" class="mw-redirect" title="McGraw-Hill Science/Engineering/Math">McGraw-Hill Science/Engineering/Math</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-054236-5" title="Special:BookSources/978-0-07-054236-5"><bdi>978-0-07-054236-5</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/21163277">21163277</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functional+Analysis&rft.place=New+York%2C+NY&rft.series=International+Series+in+Pure+and+Applied+Mathematics&rft.edition=Second&rft.pub=McGraw-Hill+Science%2FEngineering%2FMath&rft.date=1991&rft_id=info%3Aoclcnum%2F21163277&rft.isbn=978-0-07-054236-5&rft.aulast=Rudin&rft.aufirst=Walter&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffunctionalanalys00rudi&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALp+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudin1987" class="citation cs2"><a href="/wiki/Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1987), Real and complex analysis (3rd ed.), New York: <a href="/wiki/McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-054234-1" title="Special:BookSources/978-0-07-054234-1"><bdi>978-0-07-054234-1</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0924157">0924157</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+and+complex+analysis&rft.place=New+York&rft.edition=3rd&rft.pub=McGraw-Hill&rft.date=1987&rft.isbn=978-0-07-054234-1&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D924157%23id-name%3DMR&rft.aulast=Rudin&rft.aufirst=Walter&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALp+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteinShakarchi2012" class="citation book cs1">Stein, Elias M.; Shakarchi, Rami (2012). Functional Analysis: Introduction to Further Topics in Analysis. Princeton University Press. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2F9781400840557">10.1515/9781400840557</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4008-4055-7" title="Special:BookSources/978-1-4008-4055-7"><bdi>978-1-4008-4055-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functional+Analysis%3A+Introduction+to+Further+Topics+in+Analysis&rft.pub=Princeton+University+Press&rft.date=2012&rft_id=info%3Adoi%2F10.1515%2F9781400840557&rft.isbn=978-1-4008-4055-7&rft.aulast=Stein&rft.aufirst=Elias+M.&rft.au=Shakarchi%2C+Rami&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALp+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTitchmarsh1976" class="citation cs2"><a href="/wiki/Edward_Charles_Titchmarsh" title="Edward Charles Titchmarsh">Titchmarsh, EC</a> (1976), The theory of functions, Oxford University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-853349-8" title="Special:BookSources/978-0-19-853349-8"><bdi>978-0-19-853349-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+theory+of+functions&rft.pub=Oxford+University+Press&rft.date=1976&rft.isbn=978-0-19-853349-8&rft.aulast=Titchmarsh&rft.aufirst=EC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALp+space" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Lp_space&action=edit&section=32" title="Edit section: External links">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Lebesgue_space">"Lebesgue space"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Lebesgue+space&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DLebesgue_space&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALp+space" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="https://planetmath.org/ProofThatLpSpacesAreComplete">Proof that Lp spaces are complete</a></li></ul> <div class="navbox-styles"><style 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.navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Lp_spaces64" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Lp_spaces" title="Template:Lp spaces"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Lp_spaces" title="Template talk:Lp spaces"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Lp_spaces" title="Special:EditPage/Template:Lp spaces"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Lp_spaces64" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Lp spaces</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_space" title="Banach space">Banach</a> & <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a></li> <li><a class="mw-selflink selflink">Lp spaces</a></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measure</a> <ul><li><a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue</a></li></ul></li> <li><a href="/wiki/Measure_space" title="Measure space">Measure space</a></li> <li><a href="/wiki/Measurable_space" title="Measurable space">Measurable space</a>/<a href="/wiki/Measurable_function" title="Measurable function">function</a></li> <li><a href="/wiki/Minkowski_distance" title="Minkowski distance">Minkowski distance</a></li> <li><a href="/wiki/Sequence_space" title="Sequence space">Sequence spaces</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/L1_space" class="mw-redirect" title="L1 space">L1 spaces</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Integrable_function" class="mw-redirect" title="Integrable function">Integrable function</a></li> <li><a href="/wiki/Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></li> <li><a href="/wiki/Taxicab_geometry" title="Taxicab geometry">Taxicab geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/L2_space" class="mw-redirect" title="L2 space">L2 spaces</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bessel%27s_inequality" title="Bessel's inequality">Bessel's</a></li> <li><a href="/wiki/Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz</a></li> <li><a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</a></li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a></li> <li><a href="/wiki/Parseval%27s_identity" title="Parseval's identity">Parseval's identity</a></li> <li><a href="/wiki/Polarization_identity" title="Polarization identity">Polarization identity</a></li> <li><a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li> <li><a href="/wiki/Square-integrable_function" title="Square-integrable function">Square-integrable function</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/L-infinity" title="L-infinity"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ab400cc4dfd865180cd84c72dc894ca457671f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.458ex; height:2.343ex;" alt="{\displaystyle L^{\infty }}"> spaces</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bounded_function" title="Bounded function">Bounded function</a></li> <li><a href="/wiki/Chebyshev_distance" title="Chebyshev distance">Chebyshev distance</a></li> <li><a href="/wiki/Infimum_and_supremum" title="Infimum and supremum">Infimum and supremum</a> <ul><li><a href="/wiki/Essential_infimum_and_essential_supremum" title="Essential infimum and essential supremum">Essential</a></li></ul></li> <li><a href="/wiki/Uniform_norm" title="Uniform norm">Uniform norm</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Almost_everywhere" title="Almost everywhere">Almost everywhere</a></li> <li><a href="/wiki/Convergence_almost_everywhere" class="mw-redirect" title="Convergence almost everywhere">Convergence almost everywhere</a></li> <li><a href="/wiki/Convergence_in_measure" title="Convergence in measure">Convergence in measure</a></li> <li><a href="/wiki/Function_space" title="Function space">Function space</a></li> <li><a href="/wiki/Integral_transform" title="Integral transform">Integral transform</a></li> <li><a href="/wiki/Locally_integrable_function" title="Locally integrable function">Locally integrable function</a></li> <li><a href="/wiki/Measurable_function" title="Measurable function">Measurable function</a></li> <li><a href="/wiki/Symmetric_decreasing_rearrangement" title="Symmetric decreasing rearrangement">Symmetric decreasing rearrangement</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Inequalities</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Babenko%E2%80%93Beckner_inequality" title="Babenko–Beckner inequality">Babenko–Beckner</a></li> <li><a href="/wiki/Chebyshev%27s_inequality" title="Chebyshev's inequality">Chebyshev's</a></li> <li><a href="/wiki/Clarkson%27s_inequalities" title="Clarkson's inequalities">Clarkson's</a></li> <li><a href="/wiki/Hanner%27s_inequalities" title="Hanner's inequalities">Hanner's</a></li> <li><a href="/wiki/Hausdorff%E2%80%93Young_inequality" title="Hausdorff–Young inequality">Hausdorff–Young</a></li> <li><a href="/wiki/H%C3%B6lder%27s_inequality" title="Hölder's inequality">Hölder's</a></li> <li><a href="/wiki/Markov%27s_inequality" title="Markov's inequality">Markov's</a></li> <li><a href="/wiki/Minkowski_inequality" title="Minkowski inequality">Minkowski</a></li> <li><a href="/wiki/Young%27s_convolution_inequality" title="Young's convolution inequality">Young's convolution</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Theorems_in_analysis" title="Category:Theorems in analysis">Results</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Marcinkiewicz_interpolation_theorem" title="Marcinkiewicz interpolation theorem">Marcinkiewicz interpolation theorem</a></li> <li><a href="/wiki/Plancherel_theorem" title="Plancherel theorem">Plancherel theorem</a></li> <li><a href="/wiki/Riemann%E2%80%93Lebesgue_lemma" title="Riemann–Lebesgue lemma">Riemann–Lebesgue</a></li> <li><a href="/wiki/Riesz%E2%80%93Fischer_theorem" title="Riesz–Fischer theorem">Riesz–Fischer theorem</a></li> <li><a href="/wiki/Riesz%E2%80%93Thorin_theorem" title="Riesz–Thorin theorem">Riesz–Thorin theorem</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">For <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Isoperimetric_inequality" title="Isoperimetric inequality">Isoperimetric inequality</a></li> <li><a href="/wiki/Brunn%E2%80%93Minkowski_theorem" title="Brunn–Minkowski theorem">Brunn–Minkowski theorem</a> <ul><li><a href="/wiki/Milman%27s_reverse_Brunn%E2%80%93Minkowski_inequality" title="Milman's reverse Brunn–Minkowski inequality">Milman's reverse</a></li></ul></li> <li><a href="/wiki/Minkowski%E2%80%93Steiner_formula" title="Minkowski–Steiner formula">Minkowski–Steiner formula</a></li> <li><a href="/wiki/Pr%C3%A9kopa%E2%80%93Leindler_inequality" title="Prékopa–Leindler inequality">Prékopa–Leindler inequality</a></li> <li><a href="/wiki/Vitale%27s_random_Brunn%E2%80%93Minkowski_inequality" title="Vitale's random Brunn–Minkowski inequality">Vitale's random Brunn–Minkowski inequality</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications & related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bochner_space" title="Bochner space">Bochner space</a></li> <li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li> <li><a href="/wiki/Lorentz_space" title="Lorentz space">Lorentz space</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></li> <li><a href="/wiki/Quasinorm" title="Quasinorm">Quasinorm</a></li> <li><a href="/wiki/Real_analysis" title="Real analysis">Real analysis</a></li> <li><a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev space</a></li> <li><a href="/wiki/*-algebra" title="*-algebra">*-algebra</a> <ul><li><a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a></li> <li><a href="/wiki/Von_Neumann_algebra" title="Von Neumann algebra">Von Neumann</a></li></ul></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Measure_theory138" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Measure_theory" title="Template:Measure theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Measure_theory" title="Template talk:Measure theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Measure_theory" title="Special:EditPage/Template:Measure theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Measure_theory138" style="font-size:114%;margin:0 4em"><a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">Measure theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Absolute_continuity" title="Absolute continuity">Absolute continuity</a> <a href="/wiki/Absolute_continuity_(measure_theory)" class="mw-redirect" title="Absolute continuity (measure theory)">of measures</a></li> <li><a href="/wiki/Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></li> <li><a class="mw-selflink selflink">Lp spaces</a></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measure</a></li> <li><a href="/wiki/Measure_space" title="Measure space">Measure space</a> <ul><li><a href="/wiki/Probability_space" title="Probability space">Probability space</a></li></ul></li> <li><a href="/wiki/Measurable_space" title="Measurable space">Measurable space</a>/<a href="/wiki/Measurable_function" title="Measurable function">function</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sets</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Almost_everywhere" title="Almost everywhere">Almost everywhere</a></li> <li><a href="/wiki/Atom_(measure_theory)" title="Atom (measure theory)">Atom</a></li> <li><a href="/wiki/Baire_set" title="Baire set">Baire set</a></li> <li><a href="/wiki/Borel_set" title="Borel set">Borel set</a> <ul><li><a href="/wiki/Borel_equivalence_relation" title="Borel equivalence relation">equivalence relation</a></li></ul></li> <li><a href="/wiki/Standard_Borel_space" title="Standard Borel space">Borel space</a></li> <li><a href="/wiki/Carath%C3%A9odory%27s_criterion" title="Carathéodory's criterion">Carathéodory's criterion</a></li> <li><a href="/wiki/Cylindrical_%CF%83-algebra" title="Cylindrical σ-algebra">Cylindrical σ-algebra</a> <ul><li><a href="/wiki/Cylinder_set" title="Cylinder set">Cylinder set</a></li></ul></li> <li><a href="/wiki/Dynkin_system" title="Dynkin system">𝜆-system</a></li> <li><a href="/wiki/Essential_range" title="Essential range">Essential range</a> <ul><li><a href="/wiki/Essential_infimum_and_essential_supremum" title="Essential infimum and essential supremum">infimum/supremum</a></li></ul></li> <li><a href="/wiki/Locally_measurable_set" class="mw-redirect" title="Locally measurable set">Locally measurable</a></li> <li><a href="/wiki/Pi-system" title="Pi-system">π-system</a></li> <li><a href="/wiki/%CE%A3-algebra" title="Σ-algebra">σ-algebra</a></li> <li><a href="/wiki/Non-measurable_set" title="Non-measurable set">Non-measurable set</a> <ul><li><a href="/wiki/Vitali_set" title="Vitali set">Vitali set</a></li></ul></li> <li><a href="/wiki/Null_set" title="Null set">Null set</a></li> <li><a href="/wiki/Support_(measure_theory)" title="Support (measure theory)">Support</a></li> <li><a href="/wiki/Transverse_measure" title="Transverse measure">Transverse measure</a></li> <li><a href="/wiki/Universally_measurable_set" title="Universally measurable set">Universally measurable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measures</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Atomic_measure" class="mw-redirect" title="Atomic measure">Atomic</a></li> <li><a href="/wiki/Baire_measure" title="Baire measure">Baire</a></li> <li><a href="/wiki/Banach_measure" title="Banach measure">Banach</a></li> <li><a href="/wiki/Besov_measure" title="Besov measure">Besov</a></li> <li><a href="/wiki/Borel_measure" title="Borel measure">Borel</a></li> <li><a href="/wiki/Brown_measure" title="Brown measure">Brown</a></li> <li><a href="/wiki/Complex_measure" title="Complex measure">Complex</a></li> <li><a href="/wiki/Complete_measure" title="Complete measure">Complete</a></li> <li><a href="/wiki/Content_(measure_theory)" title="Content (measure theory)">Content</a></li> <li>(<a href="/wiki/Logarithmically_concave_measure" title="Logarithmically concave measure">Logarithmically</a>) <a href="/wiki/Convex_measure" title="Convex measure">Convex</a></li> <li><a href="/wiki/Decomposable_measure" title="Decomposable measure">Decomposable</a></li> <li><a href="/wiki/Discrete_measure" title="Discrete measure">Discrete</a></li> <li><a href="/wiki/Equivalence_(measure_theory)" title="Equivalence (measure theory)">Equivalent</a></li> <li><a href="/wiki/Finite_measure" title="Finite measure">Finite</a></li> <li><a href="/wiki/Inner_measure" title="Inner measure">Inner</a></li> <li>(<a href="/wiki/Quasi-invariant_measure" title="Quasi-invariant measure">Quasi-</a>) <a href="/wiki/Invariant_measure" title="Invariant measure">Invariant</a></li> <li><a href="/wiki/Locally_finite_measure" title="Locally finite measure">Locally finite</a></li> <li><a href="/wiki/Maximising_measure" title="Maximising measure">Maximising</a></li> <li><a href="/wiki/Metric_outer_measure" title="Metric outer measure">Metric outer</a></li> <li><a href="/wiki/Outer_measure" title="Outer measure">Outer</a></li> <li><a href="/wiki/Perfect_measure" title="Perfect measure">Perfect</a></li> <li><a href="/wiki/Pre-measure" title="Pre-measure">Pre-measure</a></li> <li>(<a href="/wiki/Sub-probability_measure" title="Sub-probability measure">Sub-</a>) <a href="/wiki/Probability_measure" title="Probability measure">Probability</a></li> <li><a href="/wiki/Projection-valued_measure" title="Projection-valued measure">Projection-valued</a></li> <li><a href="/wiki/Radon_measure" title="Radon measure">Radon</a></li> <li><a href="/wiki/Random_measure" title="Random measure">Random</a></li> <li><a href="/wiki/Regular_measure" title="Regular measure">Regular</a> <ul><li><a href="/wiki/Borel_regular_measure" title="Borel regular measure">Borel regular</a></li> <li><a href="/wiki/Inner_regular_measure" class="mw-redirect" title="Inner regular measure">Inner regular</a></li> <li><a href="/wiki/Outer_regular_measure" class="mw-redirect" title="Outer regular measure">Outer regular</a></li></ul></li> <li><a href="/wiki/Saturated_measure" title="Saturated measure">Saturated</a></li> <li><a href="/wiki/Set_function" title="Set function">Set function</a></li> <li><a href="/wiki/%CE%A3-finite_measure" title="Σ-finite measure">σ-finite</a></li> <li><a href="/wiki/S-finite_measure" title="S-finite measure">s-finite</a></li> <li><a href="/wiki/Signed_measure" title="Signed measure">Signed</a></li> <li><a href="/wiki/Singular_measure" title="Singular measure">Singular</a></li> <li><a href="/wiki/Spectral_measure" class="mw-redirect" title="Spectral measure">Spectral</a></li> <li><a href="/wiki/Strictly_positive_measure" title="Strictly positive measure">Strictly positive</a></li> <li><a href="/wiki/Tightness_of_measures" title="Tightness of measures">Tight</a></li> <li><a href="/wiki/Vector_measure" title="Vector measure">Vector</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Measures_(measure_theory)" title="Category:Measures (measure theory)">Particular measures</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Counting_measure" title="Counting measure">Counting</a></li> <li><a href="/wiki/Dirac_measure" title="Dirac measure">Dirac</a></li> <li><a href="/wiki/Euler_measure" title="Euler measure">Euler</a></li> <li><a href="/wiki/Gaussian_measure" title="Gaussian measure">Gaussian</a></li> <li><a href="/wiki/Haar_measure" title="Haar measure">Haar</a></li> <li><a href="/wiki/Harmonic_measure" title="Harmonic measure">Harmonic</a></li> <li><a href="/wiki/Hausdorff_measure" title="Hausdorff measure">Hausdorff</a></li> <li><a href="/wiki/Intensity_measure" title="Intensity measure">Intensity</a></li> <li><a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue</a> <ul><li><a href="/wiki/Infinite-dimensional_Lebesgue_measure" title="Infinite-dimensional Lebesgue measure">Infinite-dimensional</a></li></ul></li> <li><a href="/wiki/Positive_real_numbers#Logarithmic_measure" title="Positive real numbers">Logarithmic</a></li> <li><a href="/wiki/Product_measure" title="Product measure">Product</a> <ul><li><a href="/wiki/Projection_(measure_theory)" title="Projection (measure theory)">Projections</a></li></ul></li> <li><a href="/wiki/Pushforward_measure" title="Pushforward measure">Pushforward</a></li> <li><a href="/wiki/Spherical_measure" title="Spherical measure">Spherical measure</a></li> <li><a href="/wiki/Tangent_measure" title="Tangent measure">Tangent</a></li> <li><a href="/wiki/Trivial_measure" title="Trivial measure">Trivial</a></li> <li><a href="/wiki/Young_measure" title="Young measure">Young</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Measurable_function" title="Measurable function">Measurable function</a> <ul><li><a href="/wiki/Bochner_measurable_function" title="Bochner measurable function">Bochner</a></li> <li><a href="/wiki/Strongly_measurable_function" title="Strongly measurable function">Strongly</a></li> <li><a href="/wiki/Weakly_measurable_function" title="Weakly measurable function">Weakly</a></li></ul></li> <li>Convergence: <a href="/wiki/Convergence_almost_everywhere" class="mw-redirect" title="Convergence almost everywhere">almost everywhere</a></li> <li><a href="/wiki/Convergence_of_measures" title="Convergence of measures">of measures</a></li> <li><a href="/wiki/Convergence_in_measure" title="Convergence in measure">in measure</a></li> <li><a href="/wiki/Convergence_of_random_variables" title="Convergence of random variables">of random variables</a> <ul><li><a href="/wiki/Convergence_in_distribution" class="mw-redirect" title="Convergence in distribution">in distribution</a></li> <li><a href="/wiki/Convergence_in_probability" class="mw-redirect" title="Convergence in probability">in probability</a></li></ul></li> <li><a href="/wiki/Cylinder_set_measure" title="Cylinder set measure">Cylinder set measure</a></li> <li>Random: <a href="/wiki/Random_compact_set" title="Random compact set">compact set</a></li> <li><a href="/wiki/Random_element" title="Random element">element</a></li> <li><a href="/wiki/Random_measure" title="Random measure">measure</a></li> <li><a href="/wiki/Stochastic_process" title="Stochastic process">process</a></li> <li><a href="/wiki/Random_variable" title="Random variable">variable</a></li> <li><a href="/wiki/Multivariate_random_variable" title="Multivariate random variable">vector</a></li> <li><a href="/wiki/Projection-valued_measure" title="Projection-valued measure">Projection-valued measure</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Theorems_in_measure_theory" title="Category:Theorems in measure theory">Main results</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Carath%C3%A9odory%27s_extension_theorem" title="Carathéodory's extension theorem">Carathéodory's extension theorem</a></li> <li>Convergence theorems <ul><li><a href="/wiki/Dominated_convergence_theorem" title="Dominated convergence theorem">Dominated</a></li> <li><a href="/wiki/Monotone_convergence_theorem" title="Monotone convergence theorem">Monotone</a></li> <li><a href="/wiki/Vitali_convergence_theorem" title="Vitali convergence theorem">Vitali</a></li></ul></li> <li>Decomposition theorems <ul><li><a href="/wiki/Hahn_decomposition_theorem" title="Hahn decomposition theorem">Hahn</a></li> <li><a href="/wiki/Jordan_decomposition_theorem" class="mw-redirect" title="Jordan decomposition theorem">Jordan</a></li> <li><a href="/wiki/Maharam%27s_theorem" title="Maharam's theorem">Maharam's</a></li></ul></li> <li><a href="/wiki/Egorov%27s_theorem" title="Egorov's theorem">Egorov's</a></li> <li><a href="/wiki/Fatou%27s_lemma" title="Fatou's lemma">Fatou's lemma</a></li> <li><a href="/wiki/Fubini%27s_theorem" title="Fubini's theorem">Fubini's</a> <ul><li><a href="/wiki/Fubini%E2%80%93Tonelli_theorem" class="mw-redirect" title="Fubini–Tonelli theorem">Fubini–Tonelli</a></li></ul></li> <li><a href="/wiki/H%C3%B6lder%27s_inequality" title="Hölder's inequality">Hölder's inequality</a></li> <li><a href="/wiki/Minkowski_inequality" title="Minkowski inequality">Minkowski inequality</a></li> <li><a href="/wiki/Radon%E2%80%93Nikodym_theorem" title="Radon–Nikodym theorem">Radon–Nikodym</a></li> <li><a href="/wiki/Riesz%E2%80%93Markov%E2%80%93Kakutani_representation_theorem" title="Riesz–Markov–Kakutani representation theorem">Riesz–Markov–Kakutani representation theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other results</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Disintegration_theorem" title="Disintegration theorem">Disintegration theorem</a> <ul><li><a href="/wiki/Lifting_theory" title="Lifting theory">Lifting theory</a></li></ul></li> <li><a href="/wiki/Lebesgue%27s_density_theorem" title="Lebesgue's density theorem">Lebesgue's density theorem</a></li> <li><a href="/wiki/Lebesgue_differentiation_theorem" title="Lebesgue differentiation theorem">Lebesgue differentiation theorem</a></li> <li><a href="/wiki/Sard%27s_theorem" title="Sard's theorem">Sard's theorem</a></li> <li><a href="/wiki/Vitali%E2%80%93Hahn%E2%80%93Saks_theorem" title="Vitali–Hahn–Saks theorem">Vitali–Hahn–Saks theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">For <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Isoperimetric_inequality" title="Isoperimetric inequality">Isoperimetric inequality</a></li> <li><a href="/wiki/Brunn%E2%80%93Minkowski_theorem" title="Brunn–Minkowski theorem">Brunn–Minkowski theorem</a> <ul><li><a href="/wiki/Milman%27s_reverse_Brunn%E2%80%93Minkowski_inequality" title="Milman's reverse Brunn–Minkowski inequality">Milman's reverse</a></li></ul></li> <li><a href="/wiki/Minkowski%E2%80%93Steiner_formula" title="Minkowski–Steiner formula">Minkowski–Steiner formula</a></li> <li><a href="/wiki/Pr%C3%A9kopa%E2%80%93Leindler_inequality" title="Prékopa–Leindler inequality">Prékopa–Leindler inequality</a></li> <li><a href="/wiki/Vitale%27s_random_Brunn%E2%80%93Minkowski_inequality" title="Vitale's random Brunn–Minkowski inequality">Vitale's random Brunn–Minkowski inequality</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications & related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Convex_analysis" title="Convex analysis">Convex analysis</a></li> <li><a href="/wiki/Descriptive_set_theory" title="Descriptive set theory">Descriptive set theory</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></li> <li><a href="/wiki/Real_analysis" title="Real analysis">Real analysis</a></li> <li><a href="/wiki/Spectral_theory" title="Spectral theory">Spectral theory</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Banach_space_topics288" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Banach_spaces" title="Template:Banach spaces"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Banach_spaces" title="Template talk:Banach spaces"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Banach_spaces" title="Special:EditPage/Template:Banach spaces"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Banach_space_topics288" style="font-size:114%;margin:0 4em"><a href="/wiki/Banach_space" title="Banach space">Banach space</a> topics</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of Banach spaces</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Asplund_space" title="Asplund space">Asplund</a></li> <li><a href="/wiki/Banach_space" title="Banach space">Banach</a> <ul><li><a href="/wiki/List_of_Banach_spaces" title="List of Banach spaces">list</a></li></ul></li> <li><a href="/wiki/Banach_lattice" title="Banach lattice">Banach lattice</a></li> <li><a href="/wiki/Grothendieck_space" title="Grothendieck space">Grothendieck </a></li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert</a> <ul><li><a href="/wiki/Inner_product_space" title="Inner product space">Inner product space</a></li> <li><a href="/wiki/Polarization_identity" title="Polarization identity">Polarization identity</a></li></ul></li> <li>(<a href="/wiki/Polynomially_reflexive_space" title="Polynomially reflexive space">Polynomially</a>) <a href="/wiki/Reflexive_space" title="Reflexive space">Reflexive</a></li> <li><a href="/wiki/Riesz_space" title="Riesz space">Riesz</a></li> <li><a href="/wiki/L-semi-inner_product" title="L-semi-inner product">L-semi-inner product</a></li> <li>(<a href="/wiki/B-convex_space" title="B-convex space">B</a></li> <li><a href="/wiki/Strictly_convex_space" title="Strictly convex space">Strictly</a></li> <li><a href="/wiki/Uniformly_convex_space" title="Uniformly convex space">Uniformly</a>) convex</li> <li><a href="/wiki/Uniformly_smooth_space" title="Uniformly smooth space">Uniformly smooth</a></li> <li>(<a href="/wiki/Injective_tensor_product" title="Injective tensor product">Injective</a></li> <li><a href="/wiki/Projective_tensor_product" title="Projective tensor product">Projective</a>) <a href="/wiki/Topological_tensor_product" title="Topological tensor product">Tensor product</a> (<a href="/wiki/Tensor_product_of_Hilbert_spaces" title="Tensor product of Hilbert spaces">of Hilbert spaces</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Banach spaces are:</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Barrelled_space" title="Barrelled space">Barrelled</a></li> <li><a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">Complete</a></li> <li><a href="/wiki/F-space" title="F-space">F-space</a></li> <li><a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet</a> <ul><li><a href="/wiki/Differentiation_in_Fr%C3%A9chet_spaces#Tame_Fréchet_spaces" title="Differentiation in Fréchet spaces">tame</a></li></ul></li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex</a> <ul><li><a href="/wiki/Locally_convex_topological_vector_space#Definition_via_seminorms" title="Locally convex topological vector space">Seminorms</a>/<a href="/wiki/Minkowski_functional" title="Minkowski functional">Minkowski functionals</a></li></ul></li> <li><a href="/wiki/Mackey_space" title="Mackey space">Mackey</a></li> <li><a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">Metrizable</a></li> <li><a href="/wiki/Normed_space" class="mw-redirect" title="Normed space">Normed</a> <ul><li><a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a></li></ul></li> <li><a href="/wiki/Quasinorm" title="Quasinorm">Quasinormed</a></li> <li><a href="/wiki/Stereotype_space" class="mw-redirect" title="Stereotype space">Stereotype</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Function space Topologies</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach%E2%80%93Mazur_compactum" title="Banach–Mazur compactum">Banach–Mazur compactum</a></li> <li><a href="/wiki/Dual_topology" title="Dual topology">Dual</a></li> <li><a href="/wiki/Dual_space" title="Dual space">Dual space</a> <ul><li><a href="/wiki/Dual_norm" title="Dual norm">Dual norm</a></li></ul></li> <li><a href="/wiki/Operator_topologies" title="Operator topologies">Operator</a></li> <li><a href="/wiki/Ultraweak_topology" title="Ultraweak topology">Ultraweak</a></li> <li><a href="/wiki/Weak_topology" title="Weak topology">Weak</a> <ul><li><a href="/wiki/Weak_topology_(polar_topology)" class="mw-redirect" title="Weak topology (polar topology)">polar</a></li> <li><a href="/wiki/Weak_operator_topology" title="Weak operator topology">operator</a></li></ul></li> <li><a href="/wiki/Strong_topology" title="Strong topology">Strong</a> <ul><li><a href="/wiki/Strong_topology_(polar_topology)" class="mw-redirect" title="Strong topology (polar topology)">polar</a></li> <li><a href="/wiki/Strong_operator_topology" title="Strong operator topology">operator</a></li></ul></li> <li><a href="/wiki/Ultrastrong_topology" title="Ultrastrong topology">Ultrastrong</a></li> <li><a href="/wiki/Topology_of_uniform_convergence" class="mw-redirect" title="Topology of uniform convergence">Uniform convergence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">Linear operators</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hermitian_adjoint" title="Hermitian adjoint">Adjoint</a></li> <li><a href="/wiki/Bilinear_map" title="Bilinear map">Bilinear</a> <ul><li><a href="/wiki/Bilinear_form" title="Bilinear form">form</a></li> <li><a href="/wiki/Bilinear_map" title="Bilinear map">operator</a></li> <li><a href="/wiki/Sesquilinear_form" title="Sesquilinear form">sesquilinear</a></li></ul></li> <li>(<a href="/wiki/Unbounded_operator" title="Unbounded operator">Un</a>)<a href="/wiki/Bounded_operator" title="Bounded operator">Bounded</a></li> <li><a href="/wiki/Closed_linear_operator" title="Closed linear operator">Closed</a></li> <li><a href="/wiki/Compact_operator" title="Compact operator">Compact</a> <ul><li><a href="/wiki/Compact_operator_on_Hilbert_space" title="Compact operator on Hilbert space">on Hilbert spaces</a></li></ul></li> <li>(<a href="/wiki/Discontinuous_linear_map" title="Discontinuous linear map">Dis</a>)<a href="/wiki/Continuous_linear_operator" title="Continuous linear operator">Continuous</a></li> <li><a href="/wiki/Densely_defined" class="mw-redirect" title="Densely defined">Densely defined</a></li> <li>Fredholm <ul><li><a href="/wiki/Fredholm_kernel" title="Fredholm kernel">kernel</a></li> <li><a href="/wiki/Fredholm_operator" title="Fredholm operator">operator</a></li></ul></li> <li><a href="/wiki/Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt</a></li> <li><a href="/wiki/Linear_form" title="Linear form">Functionals</a> <ul><li><a href="/wiki/Positive_linear_functional" title="Positive linear functional">positive</a></li></ul></li> <li><a href="/wiki/Pseudo-monotone_operator" title="Pseudo-monotone operator">Pseudo-monotone</a></li> <li><a href="/wiki/Normal_operator" title="Normal operator">Normal</a></li> <li><a href="/wiki/Nuclear_operator" title="Nuclear operator">Nuclear</a></li> <li><a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">Self-adjoint</a></li> <li><a href="/wiki/Strictly_singular_operator" title="Strictly singular operator">Strictly singular</a></li> <li><a href="/wiki/Trace_class" title="Trace class">Trace class</a></li> <li><a href="/wiki/Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose</a></li> <li><a href="/wiki/Unitary_operator" title="Unitary operator">Unitary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Operator_theory" title="Operator theory">Operator theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_algebra" title="Banach algebra">Banach algebras</a></li> <li><a href="/wiki/C*-algebra" title="C*-algebra">C*-algebras</a></li> <li><a href="/wiki/Operator_space" title="Operator space">Operator space</a></li> <li><a href="/wiki/Spectrum_(functional_analysis)" title="Spectrum (functional analysis)">Spectrum</a> <ul><li><a href="/wiki/Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">C*-algebra</a></li> <li><a href="/wiki/Spectral_radius" title="Spectral radius">radius</a></li></ul></li> <li><a href="/wiki/Spectral_theory" title="Spectral theory">Spectral theory</a> <ul><li><a href="/wiki/Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">of ODEs</a></li> <li><a href="/wiki/Spectral_theorem" title="Spectral theorem">Spectral theorem</a></li></ul></li> <li><a href="/wiki/Polar_decomposition" title="Polar decomposition">Polar decomposition</a></li> <li><a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">Singular value decomposition</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Theorems_in_functional_analysis" title="Category:Theorems in functional analysis">Theorems</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Anderson%E2%80%93Kadec_theorem" title="Anderson–Kadec theorem">Anderson–Kadec</a></li> <li><a href="/wiki/Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu</a></li> <li><a href="/wiki/Banach%E2%80%93Mazur_theorem" title="Banach–Mazur theorem">Banach–Mazur</a></li> <li><a href="/wiki/Banach%E2%80%93Saks_theorem" class="mw-redirect" title="Banach–Saks theorem">Banach–Saks</a></li> <li><a href="/wiki/Open_mapping_theorem_(functional_analysis)" title="Open mapping theorem (functional analysis)">Banach–Schauder (open mapping)</a></li> <li><a href="/wiki/Uniform_boundedness_principle" title="Uniform boundedness principle">Banach–Steinhaus (Uniform boundedness)</a></li> <li><a href="/wiki/Bessel%27s_inequality" title="Bessel's inequality">Bessel's inequality</a></li> <li><a href="/wiki/Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz inequality</a></li> <li><a href="/wiki/Closed_graph_theorem" title="Closed graph theorem">Closed graph</a></li> <li><a href="/wiki/Closed_range_theorem" title="Closed range theorem">Closed range</a></li> <li><a href="/wiki/Eberlein%E2%80%93%C5%A0mulian_theorem" title="Eberlein–Šmulian theorem">Eberlein–Šmulian</a></li> <li><a href="/wiki/Freudenthal_spectral_theorem" title="Freudenthal spectral theorem">Freudenthal spectral</a></li> <li><a href="/wiki/Gelfand%E2%80%93Mazur_theorem" title="Gelfand–Mazur theorem">Gelfand–Mazur</a></li> <li><a href="/wiki/Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark</a></li> <li><a href="/wiki/Goldstine_theorem" title="Goldstine theorem">Goldstine</a></li> <li><a href="/wiki/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a> <ul><li><a href="/wiki/Hyperplane_separation_theorem" title="Hyperplane separation theorem">hyperplane separation</a></li></ul></li> <li><a href="/wiki/Kakutani_fixed-point_theorem#Infinite-dimensional_generalizations" title="Kakutani fixed-point theorem">Kakutani fixed-point</a></li> <li><a href="/wiki/Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman</a></li> <li><a href="/wiki/Invariant_subspace_problem#Known_special_cases" title="Invariant subspace problem">Lomonosov's invariant subspace</a></li> <li><a href="/wiki/Mackey%E2%80%93Arens_theorem" title="Mackey–Arens theorem">Mackey–Arens</a></li> <li><a href="/wiki/Mazur%27s_lemma" title="Mazur's lemma">Mazur's lemma</a></li> <li><a href="/wiki/M._Riesz_extension_theorem" title="M. Riesz extension theorem">M. Riesz extension</a></li> <li><a href="/wiki/Parseval%27s_identity" title="Parseval's identity">Parseval's identity</a></li> <li><a href="/wiki/Riesz%27s_lemma" title="Riesz's lemma">Riesz's lemma</a></li> <li><a href="/wiki/Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation</a></li> <li><a href="/wiki/Ursescu_theorem#Robinson–Ursescu_theorem" title="Ursescu theorem">Robinson-Ursescu</a></li> <li><a href="/wiki/Schauder_fixed-point_theorem" title="Schauder fixed-point theorem">Schauder fixed-point</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Analysis</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_Wiener_space" title="Abstract Wiener space">Abstract Wiener space</a></li> <li><a href="/wiki/Banach_manifold" title="Banach manifold">Banach manifold</a> <ul><li><a href="/wiki/Banach_bundle" title="Banach bundle">bundle</a></li></ul></li> <li><a href="/wiki/Bochner_space" title="Bochner space">Bochner space</a></li> <li><a href="/wiki/Convex_series" title="Convex series">Convex series</a></li> <li><a href="/wiki/Differentiation_in_Fr%C3%A9chet_spaces" title="Differentiation in Fréchet spaces">Differentiation in Fréchet spaces</a></li> <li><a href="/wiki/Derivative" title="Derivative">Derivatives</a> <ul><li><a href="/wiki/Fr%C3%A9chet_derivative" title="Fréchet derivative">Fréchet</a></li> <li><a href="/wiki/Gateaux_derivative" title="Gateaux derivative">Gateaux</a></li> <li><a href="/wiki/Functional_derivative" title="Functional derivative">functional</a></li> <li><a href="/wiki/Infinite-dimensional_holomorphy" title="Infinite-dimensional holomorphy">holomorphic</a></li> <li><a href="/wiki/Quasi-derivative" title="Quasi-derivative">quasi</a></li></ul></li> <li><a href="/wiki/Integral" title="Integral">Integrals</a> <ul><li><a href="/wiki/Bochner_integral" title="Bochner integral">Bochner</a></li> <li><a href="/wiki/Dunford_integral" class="mw-redirect" title="Dunford integral">Dunford</a></li> <li><a href="/wiki/Pettis_integral" title="Pettis integral">Gelfand–Pettis</a></li> <li><a href="/wiki/Regulated_integral" title="Regulated integral">regulated</a></li> <li><a href="/wiki/Paley%E2%80%93Wiener_integral" title="Paley–Wiener integral">Paley–Wiener</a></li> <li><a href="/wiki/Pettis_integral" title="Pettis integral">weak</a></li></ul></li> <li><a href="/wiki/Functional_calculus" title="Functional calculus">Functional calculus</a> <ul><li><a href="/wiki/Borel_functional_calculus" title="Borel functional calculus">Borel</a></li> <li><a href="/wiki/Continuous_functional_calculus" title="Continuous functional calculus">continuous</a></li> <li><a href="/wiki/Holomorphic_functional_calculus" title="Holomorphic functional calculus">holomorphic</a></li></ul></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measures</a> <ul><li><a href="/wiki/Infinite-dimensional_Lebesgue_measure" title="Infinite-dimensional Lebesgue measure">Lebesgue</a></li> <li><a href="/wiki/Projection-valued_measure" title="Projection-valued measure">Projection-valued</a></li> <li><a href="/wiki/Vector_measure" title="Vector measure">Vector</a></li></ul></li> <li><a href="/wiki/Weakly_measurable_function" title="Weakly measurable function">Weakly</a> / <a href="/wiki/Strongly_measurable_functions" class="mw-redirect" title="Strongly measurable functions">Strongly</a> measurable function</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of sets</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Absolutely_convex_set" title="Absolutely convex set">Absolutely convex</a></li> <li><a href="/wiki/Absorbing_set" title="Absorbing set">Absorbing</a></li> <li><a href="/wiki/Affine_space" title="Affine space">Affine</a></li> <li><a href="/wiki/Balanced_set" title="Balanced set">Balanced/Circled</a></li> <li><a href="/wiki/Bounded_set_(topological_vector_space)" title="Bounded set (topological vector space)">Bounded</a></li> <li><a href="/wiki/Convex_set" title="Convex set">Convex</a></li> <li><a href="/wiki/Convex_cone" title="Convex cone">Convex cone (subset)</a></li> <li><a href="/wiki/Convex_series#Types_of_subsets" title="Convex series">Convex series related</a> ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx))</li> <li><a href="/wiki/Cone_(linear_algebra)" class="mw-redirect" title="Cone (linear algebra)">Linear cone (subset)</a></li> <li><a href="/wiki/Radial_set" title="Radial set">Radial</a></li> <li><a href="/wiki/Star_domain" title="Star domain">Radially convex/Star-shaped</a></li> <li><a href="/wiki/Symmetric_set" title="Symmetric set">Symmetric</a></li> <li><a href="/wiki/Zonotope" class="mw-redirect" title="Zonotope">Zonotope</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Subsets / set operations</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_hull" title="Affine hull">Affine hull</a></li> <li>(<a href="/wiki/Algebraic_interior#Relative_algebraic_interior" title="Algebraic interior">Relative</a>) <a href="/wiki/Algebraic_interior" title="Algebraic interior">Algebraic interior (core)</a></li> <li><a href="/wiki/Bounding_point" title="Bounding point">Bounding points</a></li> <li><a href="/wiki/Convex_hull" title="Convex hull">Convex hull</a></li> <li><a href="/wiki/Extreme_point" title="Extreme point">Extreme point</a></li> <li><a href="/wiki/Interior_(topology)" title="Interior (topology)">Interior</a></li> <li><a href="/wiki/Linear_span" title="Linear span">Linear span</a></li> <li><a href="/wiki/Minkowski_addition" title="Minkowski addition">Minkowski addition</a></li> <li><a href="/wiki/Polar_set" title="Polar set">Polar</a></li> <li>(<a href="/wiki/Algebraic_interior#Quasi_relative_interior" title="Algebraic interior">Quasi</a>) <a href="/wiki/Algebraic_interior#Relative_interior" title="Algebraic interior">Relative interior</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Template:ListOfBanachSpaces" class="mw-redirect" title="Template:ListOfBanachSpaces">Examples</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Absolute_continuity" title="Absolute continuity">Absolute continuity AC</a></li> <li><a href="/wiki/Ba_space" title="Ba space"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ba(\Sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mi>a</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ba(\Sigma )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58fe61351e3531b14043fa2d09e98c2437bd1a6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.715ex; height:2.843ex;" alt="{\displaystyle ba(\Sigma )}"></a></li> <li><a href="/wiki/C_space" title="C space">c space</a></li> <li><a href="/wiki/BK-space" title="BK-space">Banach coordinate BK</a></li> <li><a href="/wiki/Besov_space" title="Besov space">Besov <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{p,q}^{s}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{p,q}^{s}(\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9919cf78ad095c237169772d2b27a37bfbef1b75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.524ex; height:3.009ex;" alt="{\displaystyle B_{p,q}^{s}(\mathbb {R} )}"></a></li> <li><a href="/wiki/Birnbaum%E2%80%93Orlicz_space" class="mw-redirect" title="Birnbaum–Orlicz space">Birnbaum–Orlicz</a></li> <li><a href="/wiki/Bounded_variation" title="Bounded variation">Bounded variation BV</a></li> <li><a href="/wiki/Bs_space" title="Bs space">Bs space</a></li> <li><a href="/wiki/Continuous_functions_on_a_compact_Hausdorff_space" title="Continuous functions on a compact Hausdorff space">Continuous C(K) with K compact Hausdorff</a></li> <li><a href="/wiki/Hardy_space" title="Hardy space">Hardy Hp</a></li> <li><a href="/wiki/Hilbert_space#Definition" title="Hilbert space">Hilbert H</a></li> <li><a href="/wiki/Morrey%E2%80%93Campanato_space" title="Morrey–Campanato space">Morrey–Campanato <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\lambda ,p}(\Omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo>,</mo> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\lambda ,p}(\Omega )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b8af58fa038369c3ec6386c6656aab82825e372" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.545ex; height:3.176ex;" alt="{\displaystyle L^{\lambda ,p}(\Omega )}"></a></li> <li><a href="/wiki/Sequence_space#ℓp_spaces" title="Sequence space">ℓp</a> <ul><li><a href="/wiki/L-infinity#Sequence_space" title="L-infinity"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8348195cf09473662c6f59e6717722a6fc01d0f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.845ex; height:2.343ex;" alt="{\displaystyle \ell ^{\infty }}"></a></li></ul></li> <li><a class="mw-selflink selflink">Lp</a> <ul><li><a href="/wiki/L-infinity#Function_space" title="L-infinity"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ab400cc4dfd865180cd84c72dc894ca457671f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.458ex; height:2.343ex;" alt="{\displaystyle L^{\infty }}"></a></li> <li><a class="mw-selflink-fragment" href="#Weighted_Lp_spaces">weighted</a></li></ul></li> <li><a href="/wiki/Schwartz_space" title="Schwartz space">Schwartz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\left(\mathbb {R} ^{n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\left(\mathbb {R} ^{n}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0465acd58a0f31e32b095aed742d9ccc6331369c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.592ex; height:2.843ex;" alt="{\displaystyle S\left(\mathbb {R} ^{n}\right)}"></a></li> <li><a href="/wiki/Segal%E2%80%93Bargmann_space" title="Segal–Bargmann space">Segal–Bargmann F</a></li> <li><a href="/wiki/Sequence_space" title="Sequence space">Sequence space</a></li> <li><a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev Wk,p</a> <ul><li><a href="/wiki/Sobolev_inequality" title="Sobolev inequality">Sobolev inequality</a></li></ul></li> <li><a href="/wiki/Triebel%E2%80%93Lizorkin_space" title="Triebel–Lizorkin space">Triebel–Lizorkin</a></li> <li><a href="/wiki/Wiener_amalgam_space" title="Wiener amalgam space">Wiener amalgam <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(X,L^{p})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W(X,L^{p})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b37b1dc9714960c525cb561a4828f41feb5844ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.9ex; height:2.843ex;" alt="{\displaystyle W(X,L^{p})}"></a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differential_operator" title="Differential operator">Differential operator</a></li> <li><a href="/wiki/Finite_element_method" title="Finite element method">Finite element method</a></li> <li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Mathematical formulation of quantum mechanics</a></li> <li><a href="/wiki/Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Ordinary Differential Equations (ODEs)</a></li> <li><a href="/wiki/Validated_numerics" title="Validated numerics">Validated numerics</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Functional_analysis_(topics_–_glossary)364" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Functional_analysis" title="Template:Functional analysis"><abbr title="View this 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class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_space" title="Banach space">Banach</a></li> <li><a href="/wiki/Besov_space" title="Besov space">Besov</a></li> <li><a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet</a></li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert</a></li> <li><a href="/wiki/H%C3%B6lder_space" class="mw-redirect" title="Hölder space">Hölder</a></li> <li><a href="/wiki/Nuclear_space" title="Nuclear space">Nuclear</a></li> <li><a href="/wiki/Orlicz_space" title="Orlicz space">Orlicz</a></li> <li><a href="/wiki/Schwartz_space" title="Schwartz space">Schwartz</a></li> <li><a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev</a></li> <li><a href="/wiki/Topological_vector_space" title="Topological vector space">Topological vector</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Barrelled_space" title="Barrelled space">Barrelled</a></li> <li><a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">Complete</a></li> <li><a href="/wiki/Dual_space" title="Dual space">Dual</a> (<a href="/wiki/Dual_space#Algebraic_dual_space" title="Dual space">Algebraic</a> / <a href="/wiki/Dual_space#Continuous_dual_space" title="Dual space">Topological</a>)</li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex</a></li> <li><a href="/wiki/Reflexive_space" title="Reflexive space">Reflexive</a></li> <li><a href="/wiki/Separable_space" title="Separable space">Separable</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Theorems_in_functional_analysis" title="Category:Theorems in functional analysis">Theorems</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a></li> <li><a href="/wiki/Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation</a></li> <li><a href="/wiki/Closed_graph_theorem_(functional_analysis)" title="Closed graph theorem (functional analysis)">Closed graph</a></li> <li><a href="/wiki/Uniform_boundedness_principle" title="Uniform boundedness principle">Uniform boundedness principle</a></li> <li><a href="/wiki/Kakutani_fixed-point_theorem#Infinite-dimensional_generalizations" title="Kakutani fixed-point theorem">Kakutani fixed-point</a></li> <li><a href="/wiki/Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman</a></li> <li><a href="/wiki/Min-max_theorem" title="Min-max theorem">Min–max</a></li> <li><a href="/wiki/Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark</a></li> <li><a href="/wiki/Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Operators</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adjoint_operator" class="mw-redirect" title="Adjoint operator">Adjoint</a></li> <li><a href="/wiki/Bounded_operator" title="Bounded operator">Bounded</a></li> <li><a href="/wiki/Compact_operator" title="Compact operator">Compact</a></li> <li><a href="/wiki/Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt</a></li> <li><a href="/wiki/Normal_operator" title="Normal operator">Normal</a></li> <li><a href="/wiki/Nuclear_operator" title="Nuclear operator">Nuclear</a></li> <li><a href="/wiki/Trace_class" title="Trace class">Trace class</a></li> <li><a href="/wiki/Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose</a></li> <li><a href="/wiki/Unbounded_operator" title="Unbounded operator">Unbounded</a></li> <li><a href="/wiki/Unitary_operator" title="Unitary operator">Unitary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebras</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_algebra" title="Banach algebra">Banach algebra</a></li> <li><a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a></li> <li><a href="/wiki/Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">Spectrum of a C*-algebra</a></li> <li><a href="/wiki/Operator_algebra" title="Operator algebra">Operator algebra</a></li> <li><a href="/wiki/Group_algebra_of_a_locally_compact_group" title="Group algebra of a locally compact group">Group algebra of a locally compact group</a></li> <li><a href="/wiki/Von_Neumann_algebra" title="Von Neumann algebra">Von Neumann algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Open problems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Invariant_subspace_problem" title="Invariant subspace problem">Invariant subspace problem</a></li> <li><a href="/wiki/Mahler%27s_conjecture" class="mw-redirect" title="Mahler's conjecture">Mahler's conjecture</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hardy_space" title="Hardy space">Hardy space</a></li> <li><a href="/wiki/Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Spectral theory of ordinary differential equations</a></li> <li><a href="/wiki/Heat_kernel" title="Heat kernel">Heat kernel</a></li> <li><a href="/wiki/Index_theorem" class="mw-redirect" title="Index theorem">Index theorem</a></li> <li><a href="/wiki/Calculus_of_variations" title="Calculus of variations">Calculus of variations</a></li> <li><a href="/wiki/Functional_calculus" title="Functional calculus">Functional calculus</a></li> <li><a href="/wiki/Integral_linear_operator" title="Integral linear operator">Integral linear operator</a></li> <li><a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones polynomial</a></li> <li><a href="/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">Topological quantum field theory</a></li> <li><a href="/wiki/Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a></li> <li><a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></li> <li><a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">Distribution</a> (or <a href="/wiki/Generalized_function" title="Generalized function">Generalized functions</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Advanced topics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Approximation_property" title="Approximation property">Approximation property</a></li> <li><a href="/wiki/Balanced_set" title="Balanced set">Balanced set</a></li> <li><a href="/wiki/Choquet_theory" title="Choquet theory">Choquet theory</a></li> <li><a href="/wiki/Weak_topology" title="Weak topology">Weak topology</a></li> <li><a href="/wiki/Banach%E2%80%93Mazur_distance" class="mw-redirect" title="Banach–Mazur distance">Banach–Mazur 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Lp space - Wikipedia