Inner product space

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class="vector-toc-list"> <li id="toc-Basic_properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Basic_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Basic properties</span> </div> </a> <ul id="toc-Basic_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notation"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Notation</span> </div> </a> <ul id="toc-Notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Convention_variant" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Convention_variant"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Convention variant</span> </div> </a> <ul id="toc-Convention_variant-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Examples</span> </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Real_and_complex_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Real_and_complex_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Real and complex numbers</span> </div> </a> <ul id="toc-Real_and_complex_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Euclidean_vector_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Euclidean_vector_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Euclidean vector space</span> </div> </a> <ul id="toc-Euclidean_vector_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_coordinate_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_coordinate_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Complex coordinate space</span> </div> </a> <ul id="toc-Complex_coordinate_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hilbert_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hilbert_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Hilbert space</span> </div> </a> <ul id="toc-Hilbert_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Random_variables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Random_variables"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Random variables</span> </div> </a> <ul id="toc-Random_variables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_matrices" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_matrices"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Complex matrices</span> </div> </a> <ul id="toc-Complex_matrices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vector_spaces_with_forms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vector_spaces_with_forms"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>Vector spaces with forms</span> </div> </a> <ul id="toc-Vector_spaces_with_forms-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Basic_results,_terminology,_and_definitions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Basic_results,_terminology,_and_definitions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Basic results, terminology, and definitions</span> </div> </a> <button aria-controls="toc-Basic_results,_terminology,_and_definitions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Basic results, terminology, and definitions subsection</span> </button> <ul id="toc-Basic_results,_terminology,_and_definitions-sublist" class="vector-toc-list"> <li id="toc-Norm_properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Norm_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Norm properties</span> </div> </a> <ul id="toc-Norm_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orthogonality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orthogonality"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Orthogonality</span> </div> </a> <ul id="toc-Orthogonality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Real_and_complex_parts_of_inner_products" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Real_and_complex_parts_of_inner_products"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Real and complex parts of inner products</span> </div> </a> <ul id="toc-Real_and_complex_parts_of_inner_products-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Orthonormal_sequences" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Orthonormal_sequences"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Orthonormal sequences</span> </div> </a> <ul id="toc-Orthonormal_sequences-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Operators_on_inner_product_spaces" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Operators_on_inner_product_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Operators on inner product spaces</span> </div> </a> <ul id="toc-Operators_on_inner_product_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Generalizations</span> </div> </a> <button aria-controls="toc-Generalizations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Generalizations subsection</span> </button> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> <li id="toc-Degenerate_inner_products" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Degenerate_inner_products"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Degenerate inner products</span> </div> </a> <ul id="toc-Degenerate_inner_products-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nondegenerate_conjugate_symmetric_forms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nondegenerate_conjugate_symmetric_forms"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Nondegenerate conjugate symmetric forms</span> </div> </a> <ul id="toc-Nondegenerate_conjugate_symmetric_forms-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Related_products" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_products"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Related products</span> </div> </a> <ul id="toc-Related_products-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Notes</span> </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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-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 table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Inner product space</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 29 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-29" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">29 languages</span> </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%A7%D9%84%D8%AC%D8%AF%D8%A7%D8%A1_%D8%A7%D9%84%D8%AF%D8%A7%D8%AE%D9%84%D9%8A" title="فضاء الجداء الداخلي – Arabic" lang="ar" hreflang="ar" data-title="فضاء الجداء الداخلي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Espai_prehilberti%C3%A0" title="Espai prehilbertià – Catalan" lang="ca" hreflang="ca" data-title="Espai prehilbertià" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Unit%C3%A1rn%C3%AD_prostor" title="Unitární prostor – Czech" lang="cs" hreflang="cs" data-title="Unitární prostor" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Pr%C3%A4hilbertraum" title="Prähilbertraum – German" lang="de" hreflang="de" data-title="Prähilbertraum" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Espacio_prehilbertiano" title="Espacio prehilbertiano – Spanish" lang="es" hreflang="es" data-title="Espacio prehilbertiano" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></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_%D8%B6%D8%B1%D8%A8_%D8%AF%D8%A7%D8%AE%D9%84%DB%8C" title="فضای ضرب داخلی – Persian" lang="fa" hreflang="fa" data-title="فضای ضرب داخلی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Espace_pr%C3%A9hilbertien" title="Espace préhilbertien – French" lang="fr" hreflang="fr" data-title="Espace préhilbertien" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%82%B4%EC%A0%81_%EA%B3%B5%EA%B0%84" title="내적 공간 – Korean" lang="ko" hreflang="ko" data-title="내적 공간" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Innfeldi" title="Innfeldi – Icelandic" lang="is" hreflang="is" data-title="Innfeldi" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Spazio_prehilbertiano" title="Spazio prehilbertiano – Italian" lang="it" hreflang="it" data-title="Spazio prehilbertiano" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></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_%D7%9E%D7%9B%D7%A4%D7%9C%D7%94_%D7%A4%D7%A0%D7%99%D7%9E%D7%99%D7%AA" title="מרחב מכפלה פנימית – Hebrew" lang="he" hreflang="he" data-title="מרחב מכפלה פנימית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Skal%C3%A1rszorzatos_vektort%C3%A9r" title="Skalárszorzatos vektortér – Hungarian" lang="hu" hreflang="hu" data-title="Skalárszorzatos vektortér" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Inwendig-productruimte" title="Inwendig-productruimte – Dutch" lang="nl" hreflang="nl" data-title="Inwendig-productruimte" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%A8%88%E9%87%8F%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB%E7%A9%BA%E9%96%93" title="計量ベクトル空間 – Japanese" lang="ja" hreflang="ja" data-title="計量ベクトル空間" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Indreproduktrom" title="Indreproduktrom – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Indreproduktrom" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przestrze%C5%84_unitarna" title="Przestrzeń unitarna – Polish" lang="pl" hreflang="pl" data-title="Przestrzeń unitarna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Spa%C8%9Biu_prehilbertian" title="Spațiu prehilbertian – Romanian" lang="ro" hreflang="ro" data-title="Spațiu prehilbertian" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%B5%D0%B4%D0%B3%D0%B8%D0%BB%D1%8C%D0%B1%D0%B5%D1%80%D1%82%D0%BE%D0%B2%D0%BE_%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="Предгильбертово пространство – Russian" lang="ru" hreflang="ru" data-title="Предгильбертово пространство" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Unit%C3%A1rny_priestor" title="Unitárny priestor – Slovak" lang="sk" hreflang="sk" data-title="Unitárny priestor" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Prehilbertov_prostor" title="Prehilbertov prostor – Slovenian" lang="sl" hreflang="sl" data-title="Prehilbertov prostor" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Sis%C3%A4tuloavaruus" title="Sisätuloavaruus – Finnish" lang="fi" hreflang="fi" data-title="Sisätuloavaruus" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Produktong_panloob" title="Produktong panloob – Tagalog" lang="tl" hreflang="tl" data-title="Produktong panloob" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-uk badge-Q70893996 mw-list-item" title=""><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%B5%D1%80%D0%B5%D0%B4%D0%B3%D1%96%D0%BB%D1%8C%D0%B1%D0%B5%D1%80%D1%82%D1%96%D0%B2_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%96%D1%80" title="Передгільбертів простір – Ukrainian" lang="uk" hreflang="uk" data-title="Передгільбертів простір" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%A7%D9%86%D8%AF%D8%B1%D9%88%D9%86%DB%8C_%D8%AD%D8%A7%D8%B5%D9%84_%D8%B6%D8%B1%D8%A8_%D9%81%D8%B6%D8%A7" title="اندرونی حاصل ضرب فضا – Urdu" lang="ur" hreflang="ur" data-title="اندرونی حاصل ضرب فضا" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Kh%C3%B4ng_gian_t%C3%ADch_trong" title="Không gian tích trong – 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data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Generalization of the dot product; used to define Hilbert spaces</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">"Inner product" redirects here. For the inner product of coordinate vectors, see <a href="/wiki/Dot_product" title="Dot product">Dot product</a>.</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Inner-product-angle.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Inner-product-angle.png/300px-Inner-product-angle.png" decoding="async" width="300" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Inner-product-angle.png/450px-Inner-product-angle.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Inner-product-angle.png/600px-Inner-product-angle.png 2x" data-file-width="2606" data-file-height="1995" /></a><figcaption>Geometric interpretation of the angle between two vectors defined using an inner product</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Product_Spaces_Drawing_(1).png" class="mw-file-description"><img alt="Scalar product spaces, inner product spaces, Hermitian product spaces." src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Product_Spaces_Drawing_%281%29.png/300px-Product_Spaces_Drawing_%281%29.png" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Product_Spaces_Drawing_%281%29.png/450px-Product_Spaces_Drawing_%281%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Product_Spaces_Drawing_%281%29.png/600px-Product_Spaces_Drawing_%281%29.png 2x" data-file-width="960" data-file-height="720" /></a><figcaption>Scalar product spaces, over any field, have "scalar products" that are symmetrical and linear in the first argument. Hermitian product spaces are restricted to the field of complex numbers and have "Hermitian products" that are conjugate-symmetrical and linear in the first argument. Inner product spaces may be defined over any field, having "inner products" that are linear in the first argument, conjugate-symmetrical, and positive-definite. Unlike inner products, scalar products and Hermitian products need not be positive-definite.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, an <b>inner product space</b> (or, rarely, a <b><a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a> pre-Hilbert space</b><sup id="cite_ref-FOOTNOTETrèves2006112–125_1-0" class="reference"><a href="#cite_note-FOOTNOTETrèves2006112–125-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTESchaeferWolff199940–45_2-0" class="reference"><a href="#cite_note-FOOTNOTESchaeferWolff199940–45-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>) is a <a href="/wiki/Real_vector_space" class="mw-redirect" title="Real vector space">real vector space</a> or a <a href="/wiki/Complex_vector_space" class="mw-redirect" title="Complex vector space">complex vector space</a> with an <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operation</a> called an <b>inner product</b>. The inner product of two vectors in the space is a <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a>, often denoted with <a href="/wiki/Angle_brackets" class="mw-redirect" title="Angle brackets">angle brackets</a> such as in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle a,b\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle a,b\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f9c6fc93d2e4195b610bc8a9ff366be4485387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle \langle a,b\rangle }"></span>. Inner products allow formal definitions of intuitive geometric notions, such as lengths, <a href="/wiki/Angle" title="Angle">angles</a>, and <a href="/wiki/Orthogonality" title="Orthogonality">orthogonality</a> (zero inner product) of vectors. Inner product spaces generalize <a href="/wiki/Euclidean_vector_space" class="mw-redirect" title="Euclidean vector space">Euclidean vector spaces</a>, in which the inner product is the <a href="/wiki/Dot_product" title="Dot product">dot product</a> or <i>scalar product</i> of <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a>. Inner product spaces of infinite <a href="/wiki/Dimension_(vector_space)" title="Dimension (vector space)">dimension</a> are widely used in <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>. Inner product spaces over the <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> of <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> are sometimes referred to as <b>unitary spaces</b>. The first usage of the concept of a vector space with an inner product is due to <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a>, in 1898.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>An inner product naturally induces an associated <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a>, (denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 |x|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eb41e5fd5dc37eaa1718dfbf4bc082edb991936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.623ex; height:2.843ex;" alt="{\displaystyle |x|}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |y|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |y|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee62f295ed94da43aabf12b01ce0995b83a70932" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.449ex; height:2.843ex;" alt="{\displaystyle |y|}"></span> in the picture); so, every inner product space is a <a href="/wiki/Normed_vector_space" title="Normed vector space">normed vector space</a>. If this normed space is also <a href="/wiki/Complete_metric_space" title="Complete metric space">complete</a> (that is, a <a href="/wiki/Banach_space" title="Banach space">Banach space</a>) then the inner product space is a <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a>.<sup id="cite_ref-FOOTNOTETrèves2006112–125_1-1" class="reference"><a href="#cite_note-FOOTNOTETrèves2006112–125-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> If an inner product space <span class="texhtml mvar" style="font-style:italic;">H</span> is not a Hilbert space, it can be <i>extended</i> by <a href="/wiki/Complete_topological_vector_space#Completions" title="Complete topological vector space">completion</a> to a Hilbert space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {H}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {H}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d726091334f4d8f25bd6532a6130fe74bfbe1c78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.912ex; height:3.009ex;" alt="{\displaystyle {\overline {H}}.}"></span> This means that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> is a <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspace</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {H}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {H}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0bf475c50215bef2f43a525254abbd3acc70540" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.912ex; height:3.343ex;" alt="{\displaystyle {\overline {H}},}"></span> the inner product of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> is the <a href="/wiki/Restriction_(mathematics)" title="Restriction (mathematics)">restriction</a> of that of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {H}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {H}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0bf475c50215bef2f43a525254abbd3acc70540" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.912ex; height:3.343ex;" alt="{\displaystyle {\overline {H}},}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> is <a href="/wiki/Dense_subset" class="mw-redirect" title="Dense subset">dense</a> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21e3c5eac166e464406970ed2cadc14fa7345da4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.265ex; height:3.009ex;" alt="{\displaystyle {\overline {H}}}"></span> for the <a href="/wiki/Topology_(structure)" class="mw-redirect" title="Topology (structure)">topology</a> defined by the norm.<sup id="cite_ref-FOOTNOTETrèves2006112–125_1-2" class="reference"><a href="#cite_note-FOOTNOTETrèves2006112–125-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTESchaeferWolff199936–72_4-0" class="reference"><a href="#cite_note-FOOTNOTESchaeferWolff199936–72-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In this article, <span class="texhtml"><i>F</i></span> denotes a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> that is either the <a href="/wiki/Real_number" title="Real number">real numbers</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0522388d36b55de7babe4bbfc49475eaf590c2bd" 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 \mathbb {R} ,}"></span> or the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4d5d3ec97eee8b915d3b14d3fb38579ee639d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} .}"></span> A <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a> is thus an element of <span class="texhtml"><i>F</i></span>. A bar over an expression representing a scalar denotes the <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> of this scalar. A zero vector is denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62e8c650763635a93ddc69768c3c0c100afe985d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {0} }"></span> for distinguishing it from the scalar <span class="texhtml">0</span>. </p><p>An <i>inner product</i> space is a <a href="/wiki/Vector_space" title="Vector space">vector space</a> <span class="texhtml"><i>V</i></span> over the field <span class="texhtml"><i>F</i></span> together with an <i>inner product</i>, that is, a map </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \cdot ,\cdot \rangle :V\times V\to F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mo>⋅</mo> <mo>,</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>:</mo> <mi>V</mi> <mo>×</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \cdot ,\cdot \rangle :V\times V\to F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7871e0b592252e518983a32886d33692d7b4614" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.844ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle :V\times V\to F}"></span></dd></dl> <p>that satisfies the following three properties for all vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,z\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z\in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66794d289d37b67c3857e7370ecdff292a909f34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.269ex; height:2.509ex;" alt="{\displaystyle x,y,z\in V}"></span> and all scalars <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40b0ed72d499490ba23dc89160fb8dac6664949b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.843ex; height:2.509ex;" alt="{\displaystyle a,b\in F}"></span>.<sup id="cite_ref-Jain_5-0" class="reference"><a href="#cite_note-Jain-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Prugovec̆ki_6-0" class="reference"><a href="#cite_note-Prugovec̆ki-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></span> </p> <ul><li><i>Conjugate symmetry</i>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle ={\overline {\langle y,x\rangle }}.}"> <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> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle ={\overline {\langle y,x\rangle }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786d077faef63d351ffc79e521964bffd373907d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.517ex; height:3.676ex;" alt="{\displaystyle \langle x,y\rangle ={\overline {\langle y,x\rangle }}.}"></span> As <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle a={\overline {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle a={\overline {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed015ae745f41d04b78fcdd45a0c7cf4c6ff0c2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.673ex; height:2.343ex;" alt="{\textstyle a={\overline {a}}}"></span> if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> is real, conjugate symmetry implies that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,x\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,x\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f081a4822ec7bdff2661ee2070b70a3406cf70f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.503ex; height:2.843ex;" alt="{\displaystyle \langle x,x\rangle }"></span> is always a real number. If <span class="texhtml"><i>F</i></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span>, conjugate symmetry is just symmetry.</li> <li><a href="/wiki/Linear_map" title="Linear map">Linearity</a> in the first argument:<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>Note 1<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle ax+by,z\rangle =a\langle x,z\rangle +b\langle y,z\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>a</mi> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mi>b</mi> <mo fence="false" stretchy="false">⟨</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle ax+by,z\rangle =a\langle x,z\rangle +b\langle y,z\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53bcf0b21c3515f85b53f19711663df61e1a61e1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.645ex; height:2.843ex;" alt="{\displaystyle \langle ax+by,z\rangle =a\langle x,z\rangle +b\langle y,z\rangle .}"></span></li> <li><a href="/wiki/Definite_bilinear_form" class="mw-redirect" title="Definite bilinear form">Positive-definiteness</a>: if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> is not zero, then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,x\rangle >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,x\rangle >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18c011abcbee68a18ed1fdab3307ea61c487eef2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.764ex; height:2.843ex;" alt="{\displaystyle \langle x,x\rangle >0}"></span> (conjugate symmetry implies that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,x\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,x\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f081a4822ec7bdff2661ee2070b70a3406cf70f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.503ex; height:2.843ex;" alt="{\displaystyle \langle x,x\rangle }"></span> is real).</li></ul> <p>If the positive-definiteness condition is replaced by merely requiring that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,x\rangle \geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,x\rangle \geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e6bce7d4d0f12edae272a1986cb3d93a305bdcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.764ex; height:2.843ex;" alt="{\displaystyle \langle x,x\rangle \geq 0}"></span> for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>, then one obtains the definition of <i>positive semi-definite Hermitian form</i>. A positive semi-definite Hermitian form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>⋅</mo> <mo>,</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \cdot ,\cdot \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a50080b735975d8001c9552ac2134b49ad534c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle }"></span> is an inner product if and only if for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,x\rangle =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,x\rangle =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec60641b5cb1dcfe7d2e006530aa4c8870654ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.764ex; height:2.843ex;" alt="{\displaystyle \langle x,x\rangle =0}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26c74b7807eed87581b8042b641b976798bf8edd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.765ex; height:2.176ex;" alt="{\displaystyle x=\mathbf {0} }"></span>.<sup id="cite_ref-FOOTNOTESchaeferWolff199944_8-0" class="reference"><a href="#cite_note-FOOTNOTESchaeferWolff199944-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Basic_properties">Basic properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=2" title="Edit section: Basic properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the following properties, which result almost immediately from the definition of an inner product, <span class="texhtml"><i>x</i>, <i>y</i></span> and <span class="texhtml mvar" style="font-style:italic;">z</span> are arbitrary vectors, and <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> are arbitrary scalars. </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \mathbf {0} ,x\rangle =\langle x,\mathbf {0} \rangle =0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \mathbf {0} ,x\rangle =\langle x,\mathbf {0} \rangle =0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20df841d6528e1e16a06362eeedfd74201d8295c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.025ex; height:2.843ex;" alt="{\displaystyle \langle \mathbf {0} ,x\rangle =\langle x,\mathbf {0} \rangle =0.}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,x\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,x\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f081a4822ec7bdff2661ee2070b70a3406cf70f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.503ex; height:2.843ex;" alt="{\displaystyle \langle x,x\rangle }"></span> is real and nonnegative.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,x\rangle =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,x\rangle =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec60641b5cb1dcfe7d2e006530aa4c8870654ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.764ex; height:2.843ex;" alt="{\displaystyle \langle x,x\rangle =0}"></span> if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\mathbf {0} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\mathbf {0} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9eb848d7e85fc356e69565c7061287c53c9c9d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.412ex; height:2.176ex;" alt="{\displaystyle x=\mathbf {0} .}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,ay+bz\rangle ={\overline {a}}\langle x,y\rangle +{\overline {b}}\langle x,z\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>a</mi> <mi>y</mi> <mo>+</mo> <mi>b</mi> <mi>z</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,ay+bz\rangle ={\overline {a}}\langle x,y\rangle +{\overline {b}}\langle x,z\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da5287d5d565bf475eead96ccce41568139136e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.117ex; height:3.509ex;" alt="{\displaystyle \langle x,ay+bz\rangle ={\overline {a}}\langle x,y\rangle +{\overline {b}}\langle x,z\rangle .}"></span><br />This implies that an inner product is a <a href="/wiki/Sesquilinear_form" title="Sesquilinear form">sesquilinear form</a>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x+y,x+y\rangle =\langle x,x\rangle +2\operatorname {Re} (\langle x,y\rangle )+\langle y,y\rangle ,}"> <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> <mo>,</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mn>2</mn> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>y</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x+y,x+y\rangle =\langle x,x\rangle +2\operatorname {Re} (\langle x,y\rangle )+\langle y,y\rangle ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61780c891ba11d09a2575a4503881e80a55916bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.008ex; height:2.843ex;" alt="{\displaystyle \langle x+y,x+y\rangle =\langle x,x\rangle +2\operatorname {Re} (\langle x,y\rangle )+\langle y,y\rangle ,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95843424d33425ded8dd8eb6bed645d75ebd885c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.743ex; height:2.176ex;" alt="{\displaystyle \operatorname {Re} }"></span><br />denotes the <a href="/wiki/Real_part" class="mw-redirect" title="Real part">real part</a> of its argument.</li></ul> <p>Over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span>, conjugate-symmetry reduces to symmetry, and sesquilinearity reduces to bilinearity. Hence an inner product on a real vector space is a <i>positive-definite symmetric <a href="/wiki/Bilinear_form" title="Bilinear form">bilinear form</a></i>. The <a href="/wiki/Binomial_expansion" class="mw-redirect" title="Binomial expansion">binomial expansion</a> of a square becomes </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x+y,x+y\rangle =\langle x,x\rangle +2\langle x,y\rangle +\langle y,y\rangle .}"> <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> <mo>,</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mn>2</mn> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>y</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x+y,x+y\rangle =\langle x,x\rangle +2\langle x,y\rangle +\langle y,y\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/285932a02f21f9644005947ac62ba81d1ac3f3a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.068ex; height:2.843ex;" alt="{\displaystyle \langle x+y,x+y\rangle =\langle x,x\rangle +2\langle x,y\rangle +\langle y,y\rangle .}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Notation">Notation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=3" title="Edit section: Notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Several notations are used for inner products, including <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>⋅</mo> <mo>,</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \cdot ,\cdot \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a50080b735975d8001c9552ac2134b49ad534c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\cdot ,\cdot \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mo>⋅</mo> <mo>,</mo> <mo>⋅</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\cdot ,\cdot \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4361973c49cd22e26454993b5fe8de0fb10f12a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \left(\cdot ,\cdot \right)}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \cdot |\cdot \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d36ddc87338be38721a2d721d1a6cc9ab294474a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.75ex; height:2.843ex;" alt="{\displaystyle \langle \cdot |\cdot \rangle }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\cdot |\cdot \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\cdot |\cdot \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/218bb58dc0d1c39bfc278ea3c6b064db2b053faf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.75ex; height:2.843ex;" alt="{\displaystyle \left(\cdot |\cdot \right)}"></span>, as well as the usual dot product. </p> <div class="mw-heading mw-heading3"><h3 id="Convention_variant">Convention variant</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=4" title="Edit section: Convention variant"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some authors, especially in <a href="/wiki/Physics" title="Physics">physics</a> and <a href="/wiki/Matrix_algebra" class="mw-redirect" title="Matrix algebra">matrix algebra</a>, prefer to define inner products and sesquilinear forms with linearity in the second argument rather than the first. Then the first argument becomes conjugate linear, rather than the second. <a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra-ket notation</a> in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> also uses slightly different notation, i.e. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \cdot |\cdot \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d36ddc87338be38721a2d721d1a6cc9ab294474a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.75ex; height:2.843ex;" alt="{\displaystyle \langle \cdot |\cdot \rangle }"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x|y\rangle :=\left(y,x\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>:=</mo> <mrow> <mo>(</mo> <mrow> <mi>y</mi> <mo>,</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x|y\rangle :=\left(y,x\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36b723091bb4ba6672b91683322a04f797563d72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.015ex; height:2.843ex;" alt="{\displaystyle \langle x|y\rangle :=\left(y,x\right)}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=5" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Real_and_complex_numbers">Real and complex numbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=6" title="Edit section: Real and complex numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Among the simplest examples of inner product spaces are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4d5d3ec97eee8b915d3b14d3fb38579ee639d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} .}"></span> The <a href="/wiki/Real_number" title="Real number">real numbers</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> are a vector space over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> that becomes an inner product space with arithmetic multiplication as its inner product: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle :=xy\quad {\text{ for }}x,y\in \mathbb {R} .}"> <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> <mo fence="false" stretchy="false">⟩</mo> <mo>:=</mo> <mi>x</mi> <mi>y</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle :=xy\quad {\text{ for }}x,y\in \mathbb {R} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fdf9e6d9e5ab2324ac45b65ccd173b24af9699c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.513ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle :=xy\quad {\text{ for }}x,y\in \mathbb {R} .}"></span> </p><p>The <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> are a vector space over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> that becomes an inner product space with the inner product <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle :=x{\overline {y}}\quad {\text{ for }}x,y\in \mathbb {C} .}"> <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> <mo fence="false" stretchy="false">⟩</mo> <mo>:=</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle :=x{\overline {y}}\quad {\text{ for }}x,y\in \mathbb {C} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20a0b8ba4edbba1b5fd18a85196f31db7ffd9968" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.639ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle :=x{\overline {y}}\quad {\text{ for }}x,y\in \mathbb {C} .}"></span> Unlike with the real numbers, the assignment <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)\mapsto xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)\mapsto xy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce5e1cdbb2993f6a77b27df98be18e802a126d0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.428ex; height:2.843ex;" alt="{\displaystyle (x,y)\mapsto xy}"></span> does <em>not</em> define a complex inner product on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4d5d3ec97eee8b915d3b14d3fb38579ee639d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} .}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Euclidean_vector_space">Euclidean vector space</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=7" title="Edit section: Euclidean vector space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>More generally, the <a href="/wiki/Real_coordinate_space" title="Real coordinate space">real <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>-space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}}"></span> with the <a href="/wiki/Dot_product" title="Dot product">dot product</a> is an inner product space, an example of a <a href="/wiki/Euclidean_vector_space" class="mw-redirect" title="Euclidean vector space">Euclidean vector space</a>. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle {\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}},{\begin{bmatrix}y_{1}\\\vdots \\y_{n}\end{bmatrix}}\right\rangle =x^{\textsf {T}}y=\sum _{i=1}^{n}x_{i}y_{i}=x_{1}y_{1}+\cdots +x_{n}y_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mi>y</mi> <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> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle {\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}},{\begin{bmatrix}y_{1}\\\vdots \\y_{n}\end{bmatrix}}\right\rangle =x^{\textsf {T}}y=\sum _{i=1}^{n}x_{i}y_{i}=x_{1}y_{1}+\cdots +x_{n}y_{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e87e4144c68c562a32d14a5499a1420fcca7d45" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:56.867ex; height:10.509ex;" alt="{\displaystyle \left\langle {\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}},{\begin{bmatrix}y_{1}\\\vdots \\y_{n}\end{bmatrix}}\right\rangle =x^{\textsf {T}}y=\sum _{i=1}^{n}x_{i}y_{i}=x_{1}y_{1}+\cdots +x_{n}y_{n},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\operatorname {T} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\operatorname {T} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bc69cf540c567715bca4e27c35fcd83b3a15de0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.749ex; height:2.676ex;" alt="{\displaystyle x^{\operatorname {T} }}"></span> is the <a href="/wiki/Transpose" title="Transpose">transpose</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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.}"></span> </p><p>A function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \,\cdot ,\cdot \,\rangle :\mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mspace width="thinmathspace" /> <mo>⋅</mo> <mo>,</mo> <mo>⋅</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">⟩</mo> <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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \,\cdot ,\cdot \,\rangle :\mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c12125b6ab298f0ae930f15ff8668d5b7b3140b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.774ex; height:2.843ex;" alt="{\displaystyle \langle \,\cdot ,\cdot \,\rangle :\mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} }"></span> is an inner product on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}}"></span> if and only if there exists a <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric</a> <a href="/wiki/Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">positive-definite matrix</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e499ae5946af9c09777ada933051b3669d3372c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.537ex; height:2.176ex;" alt="{\displaystyle \mathbf {M} }"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle =x^{\operatorname {T} }\mathbf {M} y}"> <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> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle =x^{\operatorname {T} }\mathbf {M} y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11f99a1cdea85a1720d667bf35f921625f765b94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.868ex; height:3.176ex;" alt="{\displaystyle \langle x,y\rangle =x^{\operatorname {T} }\mathbf {M} y}"></span> for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in \mathbb {R} ^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in \mathbb {R} ^{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56738c64d314c5aea2e35dde36597c6e2c85a203" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.903ex; height:2.676ex;" alt="{\displaystyle x,y\in \mathbb {R} ^{n}.}"></span> If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e499ae5946af9c09777ada933051b3669d3372c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.537ex; height:2.176ex;" alt="{\displaystyle \mathbf {M} }"></span> is the <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle =x^{\operatorname {T} }\mathbf {M} y}"> <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> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle =x^{\operatorname {T} }\mathbf {M} y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11f99a1cdea85a1720d667bf35f921625f765b94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.868ex; height:3.176ex;" alt="{\displaystyle \langle x,y\rangle =x^{\operatorname {T} }\mathbf {M} y}"></span> is the dot product. For another example, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a02c8bd752d2cc859747ca1f3a508281bdbc3b34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=2}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} ={\begin{bmatrix}a&b\\b&d\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mi>d</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M} ={\begin{bmatrix}a&b\\b&d\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85d9d57e6b5461d1ad396a7ad42f0926ac239a6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.611ex; height:6.176ex;" alt="{\displaystyle \mathbf {M} ={\begin{bmatrix}a&b\\b&d\end{bmatrix}}}"></span> is positive-definite (which happens if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det \mathbf {M} =ad-b^{2}>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mo>=</mo> <mi>a</mi> <mi>d</mi> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det \mathbf {M} =ad-b^{2}>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50548c121ad31d433e079f60ac964a9588ecba6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.851ex; height:2.843ex;" alt="{\displaystyle \det \mathbf {M} =ad-b^{2}>0}"></span> and one/both diagonal elements are positive) then for any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x:=\left[x_{1},x_{2}\right]^{\operatorname {T} },y:=\left[y_{1},y_{2}\right]^{\operatorname {T} }\in \mathbb {R} ^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>:=</mo> <msup> <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> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </msup> <mo>,</mo> <mi>y</mi> <mo>:=</mo> <msup> <mrow> <mo>[</mo> <mrow> <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> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </msup> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x:=\left[x_{1},x_{2}\right]^{\operatorname {T} },y:=\left[y_{1},y_{2}\right]^{\operatorname {T} }\in \mathbb {R} ^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc1e5a3f177321a76836ac0796df3c2fbddb9dde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.877ex; height:3.343ex;" alt="{\displaystyle x:=\left[x_{1},x_{2}\right]^{\operatorname {T} },y:=\left[y_{1},y_{2}\right]^{\operatorname {T} }\in \mathbb {R} ^{2},}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle :=x^{\operatorname {T} }\mathbf {M} y=\left[x_{1},x_{2}\right]{\begin{bmatrix}a&b\\b&d\end{bmatrix}}{\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}=ax_{1}y_{1}+bx_{1}y_{2}+bx_{2}y_{1}+dx_{2}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> <mo fence="false" stretchy="false">⟩</mo> <mo>:=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mi>y</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> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mi>d</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mi>a</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>b</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>b</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle :=x^{\operatorname {T} }\mathbf {M} y=\left[x_{1},x_{2}\right]{\begin{bmatrix}a&b\\b&d\end{bmatrix}}{\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}=ax_{1}y_{1}+bx_{1}y_{2}+bx_{2}y_{1}+dx_{2}y_{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0067f60aa412f08ee8a63bea3c25fec789ca61a8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:74.488ex; height:6.176ex;" alt="{\displaystyle \langle x,y\rangle :=x^{\operatorname {T} }\mathbf {M} y=\left[x_{1},x_{2}\right]{\begin{bmatrix}a&b\\b&d\end{bmatrix}}{\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}=ax_{1}y_{1}+bx_{1}y_{2}+bx_{2}y_{1}+dx_{2}y_{2}.}"></span> As mentioned earlier, every inner product on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}}"></span> is of this form (where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in \mathbb {R} ,a>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mi>a</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\in \mathbb {R} ,a>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d80b6c226eccfdf99545e90f81202ff514f8411" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.041ex; height:2.509ex;" alt="{\displaystyle b\in \mathbb {R} ,a>0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cddf6cd1242e088b641c76c3ee375466354f8d5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.477ex; height:2.176ex;" alt="{\displaystyle d>0}"></span> satisfy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ad>b^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>d</mi> <mo>></mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ad>b^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ce1cb57908a33756903fa5d3367102811a95d21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.596ex; height:2.676ex;" alt="{\displaystyle ad>b^{2}}"></span>). </p> <div class="mw-heading mw-heading3"><h3 id="Complex_coordinate_space">Complex coordinate space</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=8" title="Edit section: Complex coordinate space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The general form of an inner product on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}}"></span> is known as the <a href="/wiki/Hermitian_form" class="mw-redirect" title="Hermitian form">Hermitian form</a> and is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle =y^{\dagger }\mathbf {M} x={\overline {x^{\dagger }\mathbf {M} y}},}"> <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> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mi>y</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle =y^{\dagger }\mathbf {M} x={\overline {x^{\dagger }\mathbf {M} y}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bdf7fb19ffc1f7881b0eab1a8d75762b464aa00" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.272ex; height:4.009ex;" alt="{\displaystyle \langle x,y\rangle =y^{\dagger }\mathbf {M} x={\overline {x^{\dagger }\mathbf {M} y}},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is any <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian</a> <a href="/wiki/Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">positive-definite matrix</a> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{\dagger }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{\dagger }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1354f8c5688a03cf435a56acc9f78891a905491d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.123ex; height:3.009ex;" alt="{\displaystyle y^{\dagger }}"></span> is the <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83f72471aff7c6fbb27df0f971283a068efe091f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.802ex; height:2.009ex;" alt="{\displaystyle y.}"></span> For the real case, this corresponds to the dot product of the results of directionally-different <a href="/wiki/Scaling_(geometry)" title="Scaling (geometry)">scaling</a> of the two vectors, with positive <a href="/wiki/Scale_factor" class="mw-redirect" title="Scale factor">scale factors</a> and orthogonal directions of scaling. It is a <a href="/wiki/Weight_function" title="Weight function">weighted-sum</a> version of the dot product with positive weights—up to an orthogonal transformation. </p> <div class="mw-heading mw-heading3"><h3 id="Hilbert_space">Hilbert space</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=9" title="Edit section: Hilbert space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The article on <a href="/wiki/Hilbert_spaces" class="mw-redirect" title="Hilbert spaces">Hilbert spaces</a> has several examples of inner product spaces, wherein the metric induced by the inner product yields a <a href="/wiki/Complete_metric_space" title="Complete metric space">complete metric space</a>. An example of an inner product space which induces an incomplete metric is the space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C([a,b])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C([a,b])}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8e4f44fa2823fcdffc5fc26981c0d4fa57cade9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.131ex; height:2.843ex;" alt="{\displaystyle C([a,b])}"></span> of continuous complex valued functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> on the interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba5cb29655f824ce80a0b6a32d9326d0e8742cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.202ex; height:2.843ex;" alt="{\displaystyle [a,b].}"></span> The inner product is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle f,g\rangle =\int _{a}^{b}f(t){\overline {g(t)}}\,\mathrm {d} t.}"> <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> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</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>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle f,g\rangle =\int _{a}^{b}f(t){\overline {g(t)}}\,\mathrm {d} t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3559c0ce2344126cf72a0915602dafc6386ef041" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.099ex; height:6.343ex;" alt="{\displaystyle \langle f,g\rangle =\int _{a}^{b}f(t){\overline {g(t)}}\,\mathrm {d} t.}"></span> This space is not complete; consider for example, for the interval <span class="texhtml">[−1, 1]</span> the sequence of continuous "step" functions, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{f_{k}\}_{k},}"> <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>k</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{f_{k}\}_{k},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d855663ef5612bc74419fef776c9d80852e124a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.288ex; height:2.843ex;" alt="{\displaystyle \{f_{k}\}_{k},}"></span> defined by: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{k}(t)={\begin{cases}0&t\in [-1,0]\\1&t\in \left[{\tfrac {1}{k}},1\right]\\kt&t\in \left(0,{\tfrac {1}{k}}\right)\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo 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> <mn>0</mn> </mtd> <mtd> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi>t</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mstyle> </mrow> <mo>,</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> <mi>t</mi> </mtd> <mtd> <mi>t</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{k}(t)={\begin{cases}0&t\in [-1,0]\\1&t\in \left[{\tfrac {1}{k}},1\right]\\kt&t\in \left(0,{\tfrac {1}{k}}\right)\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e63fdb1fef334282359a93c0b232c0de8125f28" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:25.511ex; height:12.176ex;" alt="{\displaystyle f_{k}(t)={\begin{cases}0&t\in [-1,0]\\1&t\in \left[{\tfrac {1}{k}},1\right]\\kt&t\in \left(0,{\tfrac {1}{k}}\right)\end{cases}}}"></span> </p><p>This sequence is a <a href="/wiki/Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a> for the norm induced by the preceding inner product, which does not converge to a <em>continuous</em> function. </p> <div class="mw-heading mw-heading3"><h3 id="Random_variables">Random variables</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=10" title="Edit section: Random variables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For real <a href="/wiki/Random_variable" title="Random variable">random variables</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3765557b7effa1a5f2f4dce9c80a25973b7009f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.42ex; height:2.509ex;" alt="{\displaystyle Y,}"></span> the <a href="/wiki/Expected_value" title="Expected value">expected value</a> of their product <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle X,Y\rangle =\mathbb {E} [XY]}"> <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> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mi>Y</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle X,Y\rangle =\mathbb {E} [XY]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4faa9acf0f42fa31023970f64f80526fcf4ecac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.292ex; height:2.843ex;" alt="{\displaystyle \langle X,Y\rangle =\mathbb {E} [XY]}"></span> is an inner product.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> In this case, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle X,X\rangle =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>X</mi> <mo>,</mo> <mi>X</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle X,X\rangle =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c427982da938cd1340375129a47d39b94e90a9cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.064ex; height:2.843ex;" alt="{\displaystyle \langle X,X\rangle =0}"></span> if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} [X=0]=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">]</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} [X=0]=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbb889f8f5562ffa91331ceb66a78d4916a8e430" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.216ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} [X=0]=1}"></span> (that is, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d519e9e94f279ea82581dfa70a2444e896e2d860" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.241ex; height:2.176ex;" alt="{\displaystyle X=0}"></span> <a href="/wiki/Almost_surely" title="Almost surely">almost surely</a>), where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1053af9e662ceaf56c4455f90e0f67273422eded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }"></span> denotes the <a href="/wiki/Probability" title="Probability">probability</a> of the event. This definition of expectation as inner product can be extended to <a href="/wiki/Random_vector" class="mw-redirect" title="Random vector">random vectors</a> as well. </p> <div class="mw-heading mw-heading3"><h3 id="Complex_matrices">Complex matrices</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=11" title="Edit section: Complex matrices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The inner product for complex square matrices of the same size is the <a href="/wiki/Frobenius_inner_product" title="Frobenius inner product">Frobenius inner product</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle A,B\rangle :=\operatorname {tr} \left(AB^{\dagger }\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>:=</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>A</mi> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle A,B\rangle :=\operatorname {tr} \left(AB^{\dagger }\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47a18dae3ee440abe9c2fd0210f97d4b17229eeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.511ex; height:3.343ex;" alt="{\displaystyle \langle A,B\rangle :=\operatorname {tr} \left(AB^{\dagger }\right)}"></span>. Since trace and transposition are linear and the conjugation is on the second matrix, it is a sesquilinear operator. We further get Hermitian symmetry by, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle A,B\rangle =\operatorname {tr} \left(AB^{\dagger }\right)={\overline {\operatorname {tr} \left(BA^{\dagger }\right)}}={\overline {\left\langle B,A\right\rangle }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>A</mi> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>B</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mo>⟨</mo> <mrow> <mi>B</mi> <mo>,</mo> <mi>A</mi> </mrow> <mo>⟩</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle A,B\rangle =\operatorname {tr} \left(AB^{\dagger }\right)={\overline {\operatorname {tr} \left(BA^{\dagger }\right)}}={\overline {\left\langle B,A\right\rangle }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f4b554c1c3435d5ea0e11427b71bd8adab9e07c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.057ex; height:4.176ex;" alt="{\displaystyle \langle A,B\rangle =\operatorname {tr} \left(AB^{\dagger }\right)={\overline {\operatorname {tr} \left(BA^{\dagger }\right)}}={\overline {\left\langle B,A\right\rangle }}}"></span> Finally, since for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> nonzero, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle A,A\rangle =\sum _{ij}\left|A_{ij}\right|^{2}>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>A</mi> <mo>,</mo> <mi>A</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </munder> <msup> <mrow> <mo>|</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle A,A\rangle =\sum _{ij}\left|A_{ij}\right|^{2}>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3e517fcecdab0fe8518ea422decfb47d9d82626" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.999ex; height:5.843ex;" alt="{\displaystyle \langle A,A\rangle =\sum _{ij}\left|A_{ij}\right|^{2}>0}"></span>, we get that the Frobenius inner product is positive definite too, and so is an inner product. </p> <div class="mw-heading mw-heading3"><h3 id="Vector_spaces_with_forms">Vector spaces with forms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=12" title="Edit section: Vector spaces with forms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On an inner product space, or more generally a vector space with a <a href="/wiki/Nondegenerate_form" class="mw-redirect" title="Nondegenerate form">nondegenerate form</a> (hence an isomorphism <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\to V^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">→</mo> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\to V^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e86ea24eb05dd60cbcf6eb787a5f07e3861984e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.372ex; height:2.343ex;" alt="{\displaystyle V\to V^{*}}"></span>), vectors can be sent to covectors (in coordinates, via transpose), so that one can take the inner product and outer product of two vectors—not simply of a vector and a covector. </p> <div class="mw-heading mw-heading2"><h2 id="Basic_results,_terminology,_and_definitions"><span id="Basic_results.2C_terminology.2C_and_definitions"></span>Basic results, terminology, and definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=13" title="Edit section: Basic results, terminology, and definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Norm_properties">Norm properties <span class="anchor" id="Norm"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=14" title="Edit section: Norm properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every inner product space induces a <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a>, called its <em><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><span class="vanchor"><span id="canonical_norm"></span><span class="vanchor-text">canonical norm</span></span></em>, that is defined by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|={\sqrt {\langle x,x\rangle }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|={\sqrt {\langle x,x\rangle }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf3bc02e50c147bed811cfa18ee322665cb864bc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.226ex; height:4.843ex;" alt="{\displaystyle \|x\|={\sqrt {\langle x,x\rangle }}.}"></span> With this norm, every inner product space becomes a <a href="/wiki/Normed_vector_space" title="Normed vector space">normed vector space</a>. </p><p>So, every general property of normed vector spaces applies to inner product spaces. In particular, one has the following properties: </p> <style data-mw-deduplicate="TemplateStyles:r1228772891">.mw-parser-output .glossary dt{margin-top:0.4em}.mw-parser-output .glossary dt+dt{margin-top:-0.2em}.mw-parser-output .glossary .templatequote{margin-top:0;margin-bottom:-0.5em}</style> <dl class="glossary"> <dt id="absolute_homogeneity"><dfn><a href="/wiki/Absolute_homogeneity" class="mw-redirect" title="Absolute homogeneity">Absolute homogeneity</a></dfn></dt><dd> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|ax\|=|a|\,\|x\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>a</mi> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|ax\|=|a|\,\|x\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b50e6affdd645fa0292b4dc769cd51f23642bd1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.548ex; height:2.843ex;" alt="{\displaystyle \|ax\|=|a|\,\|x\|}"></span> for every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa374e20b2db7f6b8caa71ff1865f7f84f215c9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.958ex; height:2.176ex;" alt="{\displaystyle x\in V}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/130d30b75a5437ad01787d25462043ac3a9aee3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.811ex; height:2.176ex;" alt="{\displaystyle a\in F}"></span> (this results from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle ax,ax\rangle =a{\overline {a}}\langle x,x\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>a</mi> <mi>x</mi> <mo>,</mo> <mi>a</mi> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle ax,ax\rangle =a{\overline {a}}\langle x,x\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/245b21847c20211cd8d5eda19afedd24c8f1aeaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.138ex; height:2.843ex;" alt="{\displaystyle \langle ax,ax\rangle =a{\overline {a}}\langle x,x\rangle }"></span>). </dd> <dt id="triangle_inequality"><dfn><a href="/wiki/Triangle_inequality" title="Triangle inequality">Triangle inequality</a></dfn></dt><dd> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x+y\|\leq \|x\|+\|y\|}"> <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> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x+y\|\leq \|x\|+\|y\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4359c19d7e5ed2677bd29b8d46791de01b935b5b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.724ex; height:2.843ex;" alt="{\displaystyle \|x+y\|\leq \|x\|+\|y\|}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in V.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afae3d54ba364e742c2efa040fc447300966c7f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.794ex; height:2.509ex;" alt="{\displaystyle x,y\in V.}"></span> These two properties show that one has indeed a norm.</dd> <dt id="cauchy–schwarz_inequality"><dfn><a href="/wiki/Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz inequality</a></dfn></dt><dd> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\langle x,y\rangle |\leq \|x\|\,\|y\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\langle x,y\rangle |\leq \|x\|\,\|y\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f146a809cdbeebe4de138cc45d3073b26e6e9312" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.243ex; height:2.843ex;" alt="{\displaystyle |\langle x,y\rangle |\leq \|x\|\,\|y\|}"></span> for every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in V,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>V</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in V,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2958b4ac23c2fc37e5a332ac27193f757ba1452" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.794ex; height:2.509ex;" alt="{\displaystyle x,y\in V,}"></span> with equality if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> are <a href="/wiki/Linearly_independent" class="mw-redirect" title="Linearly independent">linearly dependent</a>. </dd> <dt id="parallelogram_law"><dfn><a href="/wiki/Parallelogram_law" title="Parallelogram law">Parallelogram law</a></dfn></dt><dd> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x+y\|^{2}+\|x-y\|^{2}=2\|x\|^{2}+2\|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> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x+y\|^{2}+\|x-y\|^{2}=2\|x\|^{2}+2\|y\|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7591d0aef3015a635dae6ba304875592764c3c5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.757ex; height:3.176ex;" alt="{\displaystyle \|x+y\|^{2}+\|x-y\|^{2}=2\|x\|^{2}+2\|y\|^{2}}"></span> for every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in V.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afae3d54ba364e742c2efa040fc447300966c7f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.794ex; height:2.509ex;" alt="{\displaystyle x,y\in V.}"></span> The parallelogram law is a necessary and sufficient condition for a norm to be defined by an inner product. </dd> <dt id="polarization_identity"><dfn><a href="/wiki/Polarization_identity" title="Polarization identity">Polarization identity</a></dfn></dt><dd> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x+y\|^{2}=\|x\|^{2}+\|y\|^{2}+2\operatorname {Re} \langle x,y\rangle }"> <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> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>Re</mi> <mo>⁡</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x+y\|^{2}=\|x\|^{2}+\|y\|^{2}+2\operatorname {Re} \langle x,y\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/416fb3ae3d05daeaf2bec012f6fe787bd8014705" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.348ex; height:3.176ex;" alt="{\displaystyle \|x+y\|^{2}=\|x\|^{2}+\|y\|^{2}+2\operatorname {Re} \langle x,y\rangle }"></span> for every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in V.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afae3d54ba364e742c2efa040fc447300966c7f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.794ex; height:2.509ex;" alt="{\displaystyle x,y\in V.}"></span> The inner product can be retrieved from the norm by the polarization identity, since its imaginary part is the real part of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,iy\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>i</mi> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,iy\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c557742cdf5903226465ea9e4d95ef3b53495ba2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.778ex; height:2.843ex;" alt="{\displaystyle \langle x,iy\rangle .}"></span> </dd> <dt id="ptolemy's_inequality"><dfn><a href="/wiki/Ptolemy%27s_inequality" title="Ptolemy's inequality">Ptolemy's inequality</a></dfn></dt><dd> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x-y\|\,\|z\|~+~\|y-z\|\,\|x\|~\geq ~\|x-z\|\,\|y\|}"> <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> <mo fence="false" stretchy="false">‖</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">‖</mo> <mi>z</mi> <mo fence="false" stretchy="false">‖</mo> <mtext> </mtext> <mo>+</mo> <mtext> </mtext> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo fence="false" stretchy="false">‖</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mtext> </mtext> <mo>≥</mo> <mtext> </mtext> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>z</mi> <mo fence="false" stretchy="false">‖</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x-y\|\,\|z\|~+~\|y-z\|\,\|x\|~\geq ~\|x-z\|\,\|y\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d380a0fe0fd877604f8eda3e7e53ccf178e5820" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.613ex; height:2.843ex;" alt="{\displaystyle \|x-y\|\,\|z\|~+~\|y-z\|\,\|x\|~\geq ~\|x-z\|\,\|y\|}"></span> for every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,z\in V.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z\in V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0008353bd449cdc3beba7479d1956df6e467f334" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.916ex; height:2.509ex;" alt="{\displaystyle x,y,z\in V.}"></span> Ptolemy's inequality is a necessary and sufficient condition for a <a href="/wiki/Seminorm" title="Seminorm">seminorm</a> to be the norm defined by an inner product.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </dd> </dl> <div class="mw-heading mw-heading3"><h3 id="Orthogonality">Orthogonality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=15" title="Edit section: Orthogonality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="orthogonality"><dfn><a href="/wiki/Orthogonality_(mathematics)" title="Orthogonality (mathematics)">Orthogonality</a></dfn></dt><dd> Two vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> are said to be <em><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="orthogonal"></span><span id="Orthogonal_vectors"></span><span class="vanchor-text">orthogonal</span></span></em>, often written <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\perp 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\perp y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/023639aa862e4908285cb55cd4071e430d57caa7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.23ex; height:2.509ex;" alt="{\displaystyle x\perp y,}"></span> if their inner product is zero, that is, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle =0.}"> <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> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle =0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f7176152c1a5255729e4079fbdca24b8e2f30d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.236ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle =0.}"></span> <br /> This happens if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|\leq \|x+sy\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>+</mo> <mi>s</mi> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|\leq \|x+sy\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c98bf1d9bf5270ad74505b15cea51347a6abc9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.494ex; height:2.843ex;" alt="{\displaystyle \|x\|\leq \|x+sy\|}"></span> for all scalars <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5748cdb81bf00075de8e7e6828c343687513830" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.737ex; height:2.009ex;" alt="{\displaystyle s,}"></span><sup id="cite_ref-FOOTNOTERudin1991306–312_13-0" class="reference"><a href="#cite_note-FOOTNOTERudin1991306–312-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> and if and only if the real-valued function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(s):=\|x+sy\|^{2}-\|x\|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>+</mo> <mi>s</mi> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(s):=\|x+sy\|^{2}-\|x\|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/214fc33ef1518f7a0e62fc22d771d861ac00f8d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.268ex; height:3.176ex;" alt="{\displaystyle f(s):=\|x+sy\|^{2}-\|x\|^{2}}"></span> is non-negative. (This is a consequence of the fact that, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe09d44abefff1fcada712301b088a303591f9e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.416ex; height:2.676ex;" alt="{\displaystyle y\neq 0}"></span> then the scalar <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{0}=-{\tfrac {\overline {\langle x,y\rangle }}{\|y\|^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mover> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mo accent="false">¯</mo> </mover> <mrow> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{0}=-{\tfrac {\overline {\langle x,y\rangle }}{\|y\|^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8090f7d93bf4cc52647d74af54aeada1498e83a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.475ex; height:5.676ex;" alt="{\displaystyle s_{0}=-{\tfrac {\overline {\langle x,y\rangle }}{\|y\|^{2}}}}"></span> minimizes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> with value <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(s_{0}\right)=-{\tfrac {|\langle x,y\rangle |^{2}}{\|y\|^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\left(s_{0}\right)=-{\tfrac {|\langle x,y\rangle |^{2}}{\|y\|^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a77bd79da0b489c8443d8b27497ee7c6b91ceb62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.25ex; height:5.509ex;" alt="{\displaystyle f\left(s_{0}\right)=-{\tfrac {|\langle x,y\rangle |^{2}}{\|y\|^{2}}},}"></span> which is always non positive).<br /> For a <em>complex</em> inner product space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef601e1519093ba6c2944b945882c119f990e704" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.71ex; height:2.509ex;" alt="{\displaystyle H,}"></span> a linear operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T:V\to V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T:V\to V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0fabb548ced097e03f45cf54dbb066d23d010f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.762ex; height:2.176ex;" alt="{\displaystyle T:V\to V}"></span> is identically <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\perp Tx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⊥</mo> <mi>T</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\perp Tx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b335868b131345d8f865cb00de211d2228357919" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.394ex; height:2.176ex;" alt="{\displaystyle x\perp Tx}"></span> for every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in V.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2db8a2c9b98774154e833074ffbab1d3fc6c0f61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.604ex; height:2.176ex;" alt="{\displaystyle x\in V.}"></span><sup id="cite_ref-FOOTNOTERudin1991306–312_13-1" class="reference"><a href="#cite_note-FOOTNOTERudin1991306–312-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> This is not true in general for real inner product spaces, as it is a consequence of conjugate symmetry being distinct from symmetry for complex inner products. A counterexample in a real inner product space is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> a 90° rotation in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}}"></span>, which maps every vector to an orthogonal vector but is not identically <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>. </dd> <dt id="orthogonal_complement"><dfn><a href="/wiki/Orthogonal_complement" title="Orthogonal complement">Orthogonal complement</a></dfn></dt><dd>The <i>orthogonal complement</i> of a subset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\subseteq V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>⊆</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C\subseteq V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3674daa0563e2c8ecaaa5f0fbac07010419c4269" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.652ex; height:2.343ex;" alt="{\displaystyle C\subseteq V}"></span> is the set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\bot }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊥</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{\bot }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9ba6bddf4b4b174c7ce96ac011aae08f4990689" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.309ex; height:2.676ex;" alt="{\displaystyle C^{\bot }}"></span> of the vectors that are orthogonal to all elements of <span class="texhtml mvar" style="font-style:italic;">C</span>; that is, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\bot }:=\{\,y\in V:\langle y,c\rangle =0{\text{ for all }}c\in C\,\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊥</mi> </mrow> </msup> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mspace width="thinmathspace" /> <mi>y</mi> <mo>∈</mo> <mi>V</mi> <mo>:</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>y</mi> <mo>,</mo> <mi>c</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>c</mi> <mo>∈</mo> <mi>C</mi> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{\bot }:=\{\,y\in V:\langle y,c\rangle =0{\text{ for all }}c\in C\,\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2882ec7598214b9d25588d03e99d730986cca357" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.385ex; height:3.176ex;" alt="{\displaystyle C^{\bot }:=\{\,y\in V:\langle y,c\rangle =0{\text{ for all }}c\in C\,\}.}"></span> This set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\bot }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊥</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{\bot }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9ba6bddf4b4b174c7ce96ac011aae08f4990689" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.309ex; height:2.676ex;" alt="{\displaystyle C^{\bot }}"></span> is always a closed vector subspace of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> and if the <a href="/wiki/Closure_(topology)" title="Closure (topology)">closure</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{V}C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>⁡</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{V}C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7e16ec0acae78a283477c0ec1938bbbb828693a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.329ex; height:2.509ex;" alt="{\displaystyle \operatorname {cl} _{V}C}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> is a vector subspace then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cl} _{V}C=\left(C^{\bot }\right)^{\bot }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>cl</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>⁡</mo> <mi>C</mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊥</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊥</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cl} _{V}C=\left(C^{\bot }\right)^{\bot }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/234d78f85d081e8853ec79668967f08b44119532" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.023ex; height:3.843ex;" alt="{\displaystyle \operatorname {cl} _{V}C=\left(C^{\bot }\right)^{\bot }.}"></span> </dd> <dt id="pythagorean_theorem"><dfn><a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></dfn></dt><dd> If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> are orthogonal, then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|^{2}+\|y\|^{2}=\|x+y\|^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|^{2}+\|y\|^{2}=\|x+y\|^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92c12007742388779a5f0f281c7fd58dee4cd1bb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.534ex; height:3.176ex;" alt="{\displaystyle \|x\|^{2}+\|y\|^{2}=\|x+y\|^{2}.}"></span> This may be proved by expressing the squared norms in terms of the inner products, using additivity for expanding the right-hand side of the equation.<br /> The name <em>Pythagorean theorem</em> arises from the geometric interpretation in <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>. </dd> <dt id="parseval's_identity"><dfn><a href="/wiki/Parseval%27s_identity" title="Parseval's identity">Parseval's identity</a></dfn></dt><dd> An <a href="/wiki/Mathematical_induction" title="Mathematical induction">induction</a> on the Pythagorean theorem yields: if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},\ldots ,x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},\ldots ,x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/737e02a5fbf8bc31d443c91025339f9fd1de1065" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:2.009ex;" alt="{\displaystyle x_{1},\ldots ,x_{n}}"></span> are pairwise orthogonal, then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}\|x_{i}\|^{2}=\left\|\sum _{i=1}^{n}x_{i}\right\|^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow> <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> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}\|x_{i}\|^{2}=\left\|\sum _{i=1}^{n}x_{i}\right\|^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c827e483c60b57734c04f1ca9f2318fb042e57d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.246ex; height:7.676ex;" alt="{\displaystyle \sum _{i=1}^{n}\|x_{i}\|^{2}=\left\|\sum _{i=1}^{n}x_{i}\right\|^{2}.}"></span> </dd> <dt id="angle"><dfn><a href="/wiki/Angle" title="Angle">Angle</a></dfn></dt><dd> When <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle }"> <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> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df1806ebe1fed1a728b18aed82c30be8b2a0acb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle }"></span> is a real number then the Cauchy–Schwarz inequality implies that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {\langle x,y\rangle }{\|x\|\,\|y\|}}\in [-1,1],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> <mo>∈</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\langle x,y\rangle }{\|x\|\,\|y\|}}\in [-1,1],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/971cf06efbbe8a5278a6effdacadd021c1f693a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.217ex; height:4.843ex;" alt="{\textstyle {\frac {\langle x,y\rangle }{\|x\|\,\|y\|}}\in [-1,1],}"></span> and thus that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle (x,y)=\arccos {\frac {\langle x,y\rangle }{\|x\|\,\|y\|}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle (x,y)=\arccos {\frac {\langle x,y\rangle }{\|x\|\,\|y\|}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a31c0c22827b5b760afbbadbc32886080c3ce4b3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.715ex; height:6.509ex;" alt="{\displaystyle \angle (x,y)=\arccos {\frac {\langle x,y\rangle }{\|x\|\,\|y\|}},}"></span> is a real number. This allows defining the (non oriented) <em>angle</em> of two vectors in modern definitions of <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> in terms of <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>. This is also used in <a href="/wiki/Data_analysis" title="Data analysis">data analysis</a>, under the name "<a href="/wiki/Cosine_similarity" title="Cosine similarity">cosine similarity</a>", for comparing two vectors of data. Furthermore, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle }"> <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> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df1806ebe1fed1a728b18aed82c30be8b2a0acb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle }"></span> is negative, the angle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \angle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∠</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \angle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3367b17ce7287ab814c34d00135234a07ae0228f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.006ex; height:2.843ex;" alt="{\displaystyle \angle (x,y)}"></span> is larger than 90 degrees. This property is often used in computer graphics (e.g., in <a href="/wiki/Back-face_culling" title="Back-face culling">back-face culling</a>) to analyze a direction without having to evaluate <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a>.</dd> </dl> <div class="mw-heading mw-heading3"><h3 id="Real_and_complex_parts_of_inner_products">Real and complex parts of inner products</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=16" title="Edit section: Real and complex parts of inner products"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>⋅</mo> <mo>,</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \cdot ,\cdot \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a50080b735975d8001c9552ac2134b49ad534c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle }"></span> is an inner product on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> (so it is antilinear in its second argument). The <a href="/wiki/Polarization_identity" title="Polarization identity">polarization identity</a> shows that the <a href="/wiki/Real_part" class="mw-redirect" title="Real part">real part</a> of the inner product is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} \langle x,y\rangle ={\frac {1}{4}}\left(\|x+y\|^{2}-\|x-y\|^{2}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} \langle x,y\rangle ={\frac {1}{4}}\left(\|x+y\|^{2}-\|x-y\|^{2}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4880b60107dbbd2e5611be9bbc8d68bae61e7899" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.969ex; height:5.176ex;" alt="{\displaystyle \operatorname {Re} \langle x,y\rangle ={\frac {1}{4}}\left(\|x+y\|^{2}-\|x-y\|^{2}\right).}"></span> </p><p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> is a real vector space then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle =\operatorname {Re} \langle x,y\rangle ={\frac {1}{4}}\left(\|x+y\|^{2}-\|x-y\|^{2}\right)}"> <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> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>Re</mi> <mo>⁡</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle =\operatorname {Re} \langle x,y\rangle ={\frac {1}{4}}\left(\|x+y\|^{2}-\|x-y\|^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b494e58d55a4b1fd626c64dce7e1eae9eb0502a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.362ex; height:5.176ex;" alt="{\displaystyle \langle x,y\rangle =\operatorname {Re} \langle x,y\rangle ={\frac {1}{4}}\left(\|x+y\|^{2}-\|x-y\|^{2}\right)}"></span> and the <a href="/wiki/Imaginary_part" class="mw-redirect" title="Imaginary part">imaginary part</a> (also called the <em>complex part</em>) of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>⋅</mo> <mo>,</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \cdot ,\cdot \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a50080b735975d8001c9552ac2134b49ad534c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle }"></span> is always <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/916e773e0593223c306a3e6852348177d1934962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 0.}"></span> </p><p>Assume for the rest of this section that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> is a complex vector space. The <a href="/wiki/Polarization_identity" title="Polarization identity">polarization identity</a> for complex vector spaces shows that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{4}\langle x,\ y\rangle &={\frac {1}{4}}\left(\|x+y\|^{2}-\|x-y\|^{2}+i\|x+iy\|^{2}-i\|x-iy\|^{2}\right)\\&=\operatorname {Re} \langle x,y\rangle +i\operatorname {Re} \langle x,iy\rangle .\\\end{alignedat}}}"> <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" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mtext> </mtext> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>i</mi> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>i</mi> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>i</mi> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>Re</mi> <mo>⁡</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mi>i</mi> <mi>Re</mi> <mo>⁡</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>i</mi> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{4}\langle x,\ y\rangle &={\frac {1}{4}}\left(\|x+y\|^{2}-\|x-y\|^{2}+i\|x+iy\|^{2}-i\|x-iy\|^{2}\right)\\&=\operatorname {Re} \langle x,y\rangle +i\operatorname {Re} \langle x,iy\rangle .\\\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f86bf15b338b6c4f651d6c6d3af44f8cf7b7dbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.36ex; margin-bottom: -0.312ex; width:60.824ex; height:8.509ex;" alt="{\displaystyle {\begin{alignedat}{4}\langle x,\ y\rangle &={\frac {1}{4}}\left(\|x+y\|^{2}-\|x-y\|^{2}+i\|x+iy\|^{2}-i\|x-iy\|^{2}\right)\\&=\operatorname {Re} \langle x,y\rangle +i\operatorname {Re} \langle x,iy\rangle .\\\end{alignedat}}}"></span></dd></dl> <p>The map defined by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x\mid y\rangle =\langle y,x\rangle }"> <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> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x\mid y\rangle =\langle y,x\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1a0e8e02a1536b68a985dd094870ace6dbb8947" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.659ex; height:2.843ex;" alt="{\displaystyle \langle x\mid y\rangle =\langle y,x\rangle }"></span> for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89e1c8106b3df898a0338c2ebce32902e4941fc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.147ex; height:2.509ex;" alt="{\displaystyle x,y\in V}"></span> satisfies the axioms of the inner product except that it is antilinear in its <em>first</em>, rather than its second, argument. The real part of both <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x\mid y\rangle }"> <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> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x\mid y\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5074ba53554d9c746e5ab98e1a336e5d709b5a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.232ex; height:2.843ex;" alt="{\displaystyle \langle x\mid y\rangle }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle }"> <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> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df1806ebe1fed1a728b18aed82c30be8b2a0acb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle }"></span> are equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} \langle x,y\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} \langle x,y\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90afb230cc1b685f44db36670940ea918917b302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.071ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} \langle x,y\rangle }"></span> but the inner products differ in their complex part: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{4}\langle x\mid y\rangle &={\frac {1}{4}}\left(\|x+y\|^{2}-\|x-y\|^{2}-i\|x+iy\|^{2}+i\|x-iy\|^{2}\right)\\&=\operatorname {Re} \langle x,y\rangle -i\operatorname {Re} \langle x,iy\rangle .\\\end{alignedat}}}"> <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" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>∣</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>i</mi> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>i</mi> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>i</mi> <mi>y</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>Re</mi> <mo>⁡</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>−</mo> <mi>i</mi> <mi>Re</mi> <mo>⁡</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>i</mi> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{4}\langle x\mid y\rangle &={\frac {1}{4}}\left(\|x+y\|^{2}-\|x-y\|^{2}-i\|x+iy\|^{2}+i\|x-iy\|^{2}\right)\\&=\operatorname {Re} \langle x,y\rangle -i\operatorname {Re} \langle x,iy\rangle .\\\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c231489714dbb4b6660191ac024fe34afa9e8b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.36ex; margin-bottom: -0.312ex; width:61.147ex; height:8.509ex;" alt="{\displaystyle {\begin{alignedat}{4}\langle x\mid y\rangle &={\frac {1}{4}}\left(\|x+y\|^{2}-\|x-y\|^{2}-i\|x+iy\|^{2}+i\|x-iy\|^{2}\right)\\&=\operatorname {Re} \langle x,y\rangle -i\operatorname {Re} \langle x,iy\rangle .\\\end{alignedat}}}"></span></dd></dl> <p>The last equality is similar to the formula <a href="/wiki/Real_and_imaginary_parts_of_a_linear_functional" class="mw-redirect" title="Real and imaginary parts of a linear functional">expressing a linear functional</a> in terms of its real part. </p><p>These formulas show that every complex inner product is completely determined by its real part. Moreover, this real part defines an inner product on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ace9595e3ce66fdec7e9d30202626accd676b11e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.434ex; height:2.509ex;" alt="{\displaystyle V,}"></span> considered as a real vector space. There is thus a one-to-one correspondence between complex inner products on a complex vector space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ace9595e3ce66fdec7e9d30202626accd676b11e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.434ex; height:2.509ex;" alt="{\displaystyle V,}"></span> and real inner products on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2661a49b86bd1a5548e527bbfb068aa9f59978" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.434ex; height:2.176ex;" alt="{\displaystyle V.}"></span> </p><p>For example, suppose that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <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 V=\mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c72e2d1be4153418b0ef752c7aec4dbe5f094e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.782ex; height:2.343ex;" alt="{\displaystyle V=\mathbb {C} ^{n}}"></span> for some integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57ab13b00b6f8bb19aea0dcfcc78f18a2b6cbffc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.302ex; height:2.176ex;" alt="{\displaystyle n>0.}"></span> When <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> is considered as a real vector space in the usual way (meaning that it is identified with the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n-}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>n</mi> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2n-}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4657df6e16a7ff80fb893c1e3c0a53010e330837" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.365ex; height:2.343ex;" alt="{\displaystyle 2n-}"></span>dimensional real vector space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2n},}"> <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> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/244fdd1e23c4d2f60a8da8ff339e14af86a0382d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.365ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} ^{2n},}"></span> with each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(a_{1}+ib_{1},\ldots ,a_{n}+ib_{n}\right)\in \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>∈</mo> <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 \left(a_{1}+ib_{1},\ldots ,a_{n}+ib_{n}\right)\in \mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c112c8fe5c9830ae2dfe135308d7f4f491a02138" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.011ex; height:2.843ex;" alt="{\displaystyle \left(a_{1}+ib_{1},\ldots ,a_{n}+ib_{n}\right)\in \mathbb {C} ^{n}}"></span> identified with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(a_{1},b_{1},\ldots ,a_{n},b_{n}\right)\in \mathbb {R} ^{2n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(a_{1},b_{1},\ldots ,a_{n},b_{n}\right)\in \mathbb {R} ^{2n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/539952abc4b278f097f8fc64b85801052c62e7f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.615ex; height:3.176ex;" alt="{\displaystyle \left(a_{1},b_{1},\ldots ,a_{n},b_{n}\right)\in \mathbb {R} ^{2n}}"></span>), then the <a href="/wiki/Dot_product" title="Dot product">dot product</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\,\cdot \,y=\left(x_{1},\ldots ,x_{2n}\right)\,\cdot \,\left(y_{1},\ldots ,y_{2n}\right):=x_{1}y_{1}+\cdots +x_{2n}y_{2n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mspace width="thinmathspace" /> <mo>⋅</mo> <mspace width="thinmathspace" /> <mi>y</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>⋅</mo> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>:=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\,\cdot \,y=\left(x_{1},\ldots ,x_{2n}\right)\,\cdot \,\left(y_{1},\ldots ,y_{2n}\right):=x_{1}y_{1}+\cdots +x_{2n}y_{2n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e07b04a6ba781d1ba197d0237b69479388c9da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.869ex; height:2.843ex;" alt="{\displaystyle x\,\cdot \,y=\left(x_{1},\ldots ,x_{2n}\right)\,\cdot \,\left(y_{1},\ldots ,y_{2n}\right):=x_{1}y_{1}+\cdots +x_{2n}y_{2n}}"></span> defines a real inner product on this space. The unique complex inner product <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>⋅</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \,\cdot ,\cdot \,\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd6032140d9a706feba8f8a6347597ea068a7219" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.911ex; height:2.843ex;" alt="{\displaystyle \langle \,\cdot ,\cdot \,\rangle }"></span> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <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 V=\mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c72e2d1be4153418b0ef752c7aec4dbe5f094e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.782ex; height:2.343ex;" alt="{\displaystyle V=\mathbb {C} ^{n}}"></span> induced by the dot product is the map that sends <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=\left(c_{1},\ldots ,c_{n}\right),d=\left(d_{1},\ldots ,d_{n}\right)\in \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>∈</mo> <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 c=\left(c_{1},\ldots ,c_{n}\right),d=\left(d_{1},\ldots ,d_{n}\right)\in \mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7250f3fdb0599ca8cd813bfba2d04e234eb6e4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.53ex; height:2.843ex;" alt="{\displaystyle c=\left(c_{1},\ldots ,c_{n}\right),d=\left(d_{1},\ldots ,d_{n}\right)\in \mathbb {C} ^{n}}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle c,d\rangle :=c_{1}{\overline {d_{1}}}+\cdots +c_{n}{\overline {d_{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>:=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle c,d\rangle :=c_{1}{\overline {d_{1}}}+\cdots +c_{n}{\overline {d_{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0697e270aa830b3aade33e6e2f28ab7632bd8ba0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.422ex; height:3.509ex;" alt="{\displaystyle \langle c,d\rangle :=c_{1}{\overline {d_{1}}}+\cdots +c_{n}{\overline {d_{n}}}}"></span> (because the real part of this map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>⋅</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \,\cdot ,\cdot \,\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd6032140d9a706feba8f8a6347597ea068a7219" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.911ex; height:2.843ex;" alt="{\displaystyle \langle \,\cdot ,\cdot \,\rangle }"></span> is equal to the dot product). </p><p><b>Real vs. complex inner products</b> </p><p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathbb {R} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathbb {R} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43e1ed20fc2bddc69536b4c4f85a4c5d81a62ecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.774ex; height:2.509ex;" alt="{\displaystyle V_{\mathbb {R} }}"></span> denote <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> considered as a vector space over the real numbers rather than complex numbers. The <a href="/wiki/Real_part" class="mw-redirect" title="Real part">real part</a> of the complex inner product <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle }"> <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> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df1806ebe1fed1a728b18aed82c30be8b2a0acb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle }"></span> is the map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle _{\mathbb {R} }=\operatorname {Re} \langle x,y\rangle ~:~V_{\mathbb {R} }\times V_{\mathbb {R} }\to \mathbb {R} ,}"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>Re</mi> <mo>⁡</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mtext> </mtext> <mo>:</mo> <mtext> </mtext> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mo>×</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle _{\mathbb {R} }=\operatorname {Re} \langle x,y\rangle ~:~V_{\mathbb {R} }\times V_{\mathbb {R} }\to \mathbb {R} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f6bf74f5ca9b18d42b1f5bfb50a43661c2204f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.343ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle _{\mathbb {R} }=\operatorname {Re} \langle x,y\rangle ~:~V_{\mathbb {R} }\times V_{\mathbb {R} }\to \mathbb {R} ,}"></span> which necessarily forms a real inner product on the real vector space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathbb {R} }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathbb {R} }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b45a2de7adf0eebcb8df7271dd5ad002337f3321" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.421ex; height:2.509ex;" alt="{\displaystyle V_{\mathbb {R} }.}"></span> Every inner product on a real vector space is a <a href="/wiki/Bilinear_map" title="Bilinear map">bilinear</a> and <a href="/wiki/Symmetric_map" class="mw-redirect" title="Symmetric map">symmetric map</a>. </p><p>For example, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f474512c3b57d03034c79cc18bb5ecf448c3f51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.564ex; height:2.176ex;" alt="{\displaystyle V=\mathbb {C} }"></span> with inner product <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle =x{\overline {y}},}"> <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> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle =x{\overline {y}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/098403440f3a4de380a1eaea1b212f73240f16b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.684ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle =x{\overline {y}},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> is a vector space over the field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ff6a3dc2982018ff20f1d2c927afc74a217be6" 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 \mathbb {C} ,}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathbb {R} }=\mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mo>=</mo> <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 V_{\mathbb {R} }=\mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31e3d58ce17b831e779ace3ec350cc3a76fdd54b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.605ex; height:3.009ex;" alt="{\displaystyle V_{\mathbb {R} }=\mathbb {R} ^{2}}"></span> is a vector space over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle _{\mathbb {R} }}"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle _{\mathbb {R} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3736dae913b80683b7c360f23c37830f1ed77520" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.747ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle _{\mathbb {R} }}"></span> is the <a href="/wiki/Dot_product" title="Dot product">dot product</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\cdot 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\cdot y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/699802977820dd08f3741b402f6a5d2ae097dc02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.811ex; height:2.009ex;" alt="{\displaystyle x\cdot y,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=a+ib\in V=\mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> <mo>∈</mo> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=a+ib\in V=\mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc60ff91956ac3fc7f9518a949e843cb231db436" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.703ex; height:2.343ex;" alt="{\displaystyle x=a+ib\in V=\mathbb {C} }"></span> is identified with the point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)\in V_{\mathbb {R} }=\mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mo>=</mo> <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 (a,b)\in V_{\mathbb {R} }=\mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afb7730b87d50f70f03e98c5bb8ac03c9bafe8e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.516ex; height:3.176ex;" alt="{\displaystyle (a,b)\in V_{\mathbb {R} }=\mathbb {R} ^{2}}"></span> (and similarly for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>); thus the standard inner product <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle =x{\overline {y}},}"> <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> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle =x{\overline {y}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/098403440f3a4de380a1eaea1b212f73240f16b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.684ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle =x{\overline {y}},}"></span> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span> is an "extension" the dot product . Also, had <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle }"> <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> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df1806ebe1fed1a728b18aed82c30be8b2a0acb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle }"></span> been instead defined to be the <b><a href="#math_Symmetry">symmetric map</a></b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle =xy}"> <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> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle =xy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bb0d72c852448abffb7eee62c451bcd58e7776e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.912ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle =xy}"></span> (rather than the usual <b><a href="#math_Conjugate_symmetry">conjugate symmetric map</a></b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle =x{\overline {y}}}"> <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> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle =x{\overline {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47ffd1651587afb92a776d9bc0e9d1299fe358b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.038ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle =x{\overline {y}}}"></span>) then its real part <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle _{\mathbb {R} }}"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle _{\mathbb {R} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3736dae913b80683b7c360f23c37830f1ed77520" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.747ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle _{\mathbb {R} }}"></span> would <em>not</em> be the dot product; furthermore, without the complex conjugate, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75e8f71807e571bdc2c1545cfe8eebfe26943789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle x\in \mathbb {C} }"></span> but <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\not \in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∉</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\not \in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aad1ce3995cdfc0385c46342d0403b8ceaa3136e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.848ex; height:2.676ex;" alt="{\displaystyle x\not \in \mathbb {R} }"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,x\rangle =xx=x^{2}\not \in [0,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>x</mi> <mi>x</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>∉</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 \langle x,x\rangle =xx=x^{2}\not \in [0,\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9380c0cf12043a4c60abd5c57f271a5062b4114b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.655ex; height:3.176ex;" alt="{\displaystyle \langle x,x\rangle =xx=x^{2}\not \in [0,\infty )}"></span> so the assignment <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto {\sqrt {\langle x,x\rangle }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto {\sqrt {\langle x,x\rangle }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aabe196647a5ef056910b3c95529ccf7028dd1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.77ex; height:4.843ex;" alt="{\displaystyle x\mapsto {\sqrt {\langle x,x\rangle }}}"></span> would not define a norm. </p><p>The next examples show that although real and complex inner products have many properties and results in common, they are not entirely interchangeable. For instance, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle =0}"> <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> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e39d8b3de5c9ae1f31e5a6515f1a1156c5187702" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.589ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle =0}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle _{\mathbb {R} }=0,}"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle _{\mathbb {R} }=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50744b7b65cfd568184b8c66f20b83cbf1c05de6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.655ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle _{\mathbb {R} }=0,}"></span> but the next example shows that the converse is in general <em>not</em> true. Given any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in V,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>V</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in V,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bcfd2b7752b39f2d16a9140f288e243f84c4c30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.604ex; height:2.509ex;" alt="{\displaystyle x\in V,}"></span> the vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ix}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ix}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8620946c5b679256caa25581a6fb98efa502141e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.132ex; height:2.176ex;" alt="{\displaystyle ix}"></span> (which is the vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> rotated by 90°) belongs to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> and so also belongs to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathbb {R} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathbb {R} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43e1ed20fc2bddc69536b4c4f85a4c5d81a62ecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.774ex; height:2.509ex;" alt="{\displaystyle V_{\mathbb {R} }}"></span> (although scalar multiplication of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i={\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i={\sqrt {-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/370c8cebe9634fbfc84c29ea61680b0ad4a1ae0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.807ex; height:3.009ex;" alt="{\displaystyle i={\sqrt {-1}}}"></span> is not defined in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathbb {R} },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathbb {R} },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d57c04b67cb40b5edffff9c835c3d1f1e17f73f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.421ex; height:2.509ex;" alt="{\displaystyle V_{\mathbb {R} },}"></span> the vector in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ix}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ix}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8620946c5b679256caa25581a6fb98efa502141e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.132ex; height:2.176ex;" alt="{\displaystyle ix}"></span> is nevertheless still also an element of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathbb {R} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathbb {R} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43e1ed20fc2bddc69536b4c4f85a4c5d81a62ecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.774ex; height:2.509ex;" alt="{\displaystyle V_{\mathbb {R} }}"></span>). For the complex inner product, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,ix\rangle =-i\|x\|^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>i</mi> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,ix\rangle =-i\|x\|^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59df9b1a5aa455498d693f7b31ae21addd5eac16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.37ex; height:3.176ex;" alt="{\displaystyle \langle x,ix\rangle =-i\|x\|^{2},}"></span> whereas for the real inner product the value is always <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,ix\rangle _{\mathbb {R} }=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>i</mi> <mi>x</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,ix\rangle _{\mathbb {R} }=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e6670b16315ba15bf0ee997f9216acde2a8b71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.632ex; height:2.843ex;" alt="{\displaystyle \langle x,ix\rangle _{\mathbb {R} }=0.}"></span> </p><p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>⋅</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \,\cdot ,\cdot \,\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd6032140d9a706feba8f8a6347597ea068a7219" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.911ex; height:2.843ex;" alt="{\displaystyle \langle \,\cdot ,\cdot \,\rangle }"></span> is a complex inner product and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A:V\to V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A:V\to V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2348ed6dd9b0f7cb231284ab34d02a6b60379ab8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.869ex; height:2.176ex;" alt="{\displaystyle A:V\to V}"></span> is a continuous linear operator that satisfies <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,Ax\rangle =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>A</mi> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,Ax\rangle =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f5c57c9614087f44141f07f308a86a9f8f136f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.507ex; height:2.843ex;" alt="{\displaystyle \langle x,Ax\rangle =0}"></span> for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in V,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>V</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in V,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bcfd2b7752b39f2d16a9140f288e243f84c4c30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.604ex; height:2.509ex;" alt="{\displaystyle x\in V,}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=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=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01dbe289a1d153a8232c394390712801aaf2613b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.651ex; height:2.176ex;" alt="{\displaystyle A=0.}"></span> This statement is no longer true if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>⋅</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \,\cdot ,\cdot \,\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd6032140d9a706feba8f8a6347597ea068a7219" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.911ex; height:2.843ex;" alt="{\displaystyle \langle \,\cdot ,\cdot \,\rangle }"></span> is instead a real inner product, as this next example shows. Suppose that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f474512c3b57d03034c79cc18bb5ecf448c3f51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.564ex; height:2.176ex;" alt="{\displaystyle V=\mathbb {C} }"></span> has the inner product <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle :=x{\overline {y}}}"> <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> <mo fence="false" stretchy="false">⟩</mo> <mo>:=</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle :=x{\overline {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d8fa3d11722ab4699ddde7bfaeda3c5eb076304" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.684ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle :=x{\overline {y}}}"></span> mentioned above. Then the map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A:V\to V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A:V\to V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2348ed6dd9b0f7cb231284ab34d02a6b60379ab8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.869ex; height:2.176ex;" alt="{\displaystyle A:V\to V}"></span> defined by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ax=ix}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>x</mi> <mo>=</mo> <mi>i</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax=ix}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a2be348112e904b0634bc3fec32d4431110e94e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.303ex; height:2.176ex;" alt="{\displaystyle Ax=ix}"></span> is a linear map (linear for both <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathbb {R} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathbb {R} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43e1ed20fc2bddc69536b4c4f85a4c5d81a62ecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.774ex; height:2.509ex;" alt="{\displaystyle V_{\mathbb {R} }}"></span>) that denotes rotation by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 90^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c326d317eddef3ad3e6625e018a708e290a039f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 90^{\circ }}"></span> in the plane. Because <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ax}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4527572c17e687922277b4ffe25b28c9411fc0a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.073ex; height:2.176ex;" alt="{\displaystyle Ax}"></span> are perpendicular vectors and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,Ax\rangle _{\mathbb {R} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>A</mi> <mi>x</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,Ax\rangle _{\mathbb {R} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c84d35607d2bba1ca9d815d9bc38f42c0b7d5625" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.665ex; height:2.843ex;" alt="{\displaystyle \langle x,Ax\rangle _{\mathbb {R} }}"></span> is just the dot product, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,Ax\rangle _{\mathbb {R} }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>A</mi> <mi>x</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,Ax\rangle _{\mathbb {R} }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7afcac40a427167f955b2f7c5697cccfca78788" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.925ex; height:2.843ex;" alt="{\displaystyle \langle x,Ax\rangle _{\mathbb {R} }=0}"></span> for all vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05a950b252371110b85a784de7babc2448d28cc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x;}"></span> nevertheless, this rotation map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is certainly not identically <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/916e773e0593223c306a3e6852348177d1934962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 0.}"></span> In contrast, using the complex inner product gives <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,Ax\rangle =-i\|x\|^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>A</mi> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,Ax\rangle =-i\|x\|^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d202dae8bfa576bb636ded5a6a518f766c30efe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.31ex; height:3.176ex;" alt="{\displaystyle \langle x,Ax\rangle =-i\|x\|^{2},}"></span> which (as expected) is not identically zero. </p> <div class="mw-heading mw-heading2"><h2 id="Orthonormal_sequences">Orthonormal sequences</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=17" title="Edit section: Orthonormal sequences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Orthogonal_basis" title="Orthogonal basis">Orthogonal basis</a> and <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">Orthonormal basis</a></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> be a finite dimensional inner product space of dimension <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e59df02a9f67a5da3c220f1244c99a46cc4eb1c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:1.676ex;" alt="{\displaystyle n.}"></span> Recall that every <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> consists of exactly <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> linearly independent vectors. Using the <a href="/wiki/Gram%E2%80%93Schmidt_process" title="Gram–Schmidt process">Gram–Schmidt process</a> we may start with an arbitrary basis and transform it into an orthonormal basis. That is, into a basis in which all the elements are orthogonal and have unit norm. In symbols, a basis <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{e_{1},\ldots ,e_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{e_{1},\ldots ,e_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/014c2407d0cf40698e1a1ba829d2622919d89ede" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.943ex; height:2.843ex;" alt="{\displaystyle \{e_{1},\ldots ,e_{n}\}}"></span> is orthonormal if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle e_{i},e_{j}\rangle =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle e_{i},e_{j}\rangle =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a263693d5bf1481495410877131a2bb5b5221985" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.981ex; height:3.009ex;" alt="{\displaystyle \langle e_{i},e_{j}\rangle =0}"></span> for every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\neq j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\neq j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d95aeb406bb427ac96806bc00c30c91d31b858be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.859ex; height:2.676ex;" alt="{\displaystyle i\neq j}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle e_{i},e_{i}\rangle =\|e_{a}\|^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle e_{i},e_{i}\rangle =\|e_{a}\|^{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39c0662b6833537b00daec7685b1c6cce471cfc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.533ex; height:3.176ex;" alt="{\displaystyle \langle e_{i},e_{i}\rangle =\|e_{a}\|^{2}=1}"></span> for each index <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ffcf9ad7ad44f04fa43c5b604b4801e089981cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.449ex; height:2.176ex;" alt="{\displaystyle i.}"></span> </p><p>This definition of orthonormal basis generalizes to the case of infinite-dimensional inner product spaces in the following way. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> be any inner product space. Then a collection <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\left\{e_{a}\right\}_{a\in A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <msub> <mrow> <mo>{</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>∈</mo> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=\left\{e_{a}\right\}_{a\in A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8472f13c8f7e47a04bd8faa166efd8a5fe29b759" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.815ex; height:3.009ex;" alt="{\displaystyle E=\left\{e_{a}\right\}_{a\in A}}"></span> is a <em>basis</em> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> if the subspace of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> generated by finite linear combinations of elements of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> is dense in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> (in the norm induced by the inner product). Say that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> is an <em><a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a></em> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> if it is a basis and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle e_{a},e_{b}\right\rangle =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle e_{a},e_{b}\right\rangle =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64a834bf013d73d8b5c8f6df291d78850e661931" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.311ex; height:2.843ex;" alt="{\displaystyle \left\langle e_{a},e_{b}\right\rangle =0}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\neq b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≠</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\neq b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bf17ce351d08f2b7a3e54ba3aa2132f260c84f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.326ex; height:2.676ex;" alt="{\displaystyle a\neq b}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle e_{a},e_{a}\rangle =\|e_{a}\|^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle e_{a},e_{a}\rangle =\|e_{a}\|^{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30bf5414b9553cb12f6589f0f8f569d3afb3fe8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.138ex; height:3.176ex;" alt="{\displaystyle \langle e_{a},e_{a}\rangle =\|e_{a}\|^{2}=1}"></span> for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in A.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>A</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in A.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59ffe5b62cb35c2f183efd33cc66e9d30d68cbda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.492ex; height:2.509ex;" alt="{\displaystyle a,b\in A.}"></span> </p><p>Using an infinite-dimensional analog of the Gram-Schmidt process one may show: </p><p><b>Theorem.</b> Any <a href="/wiki/Separable_space" title="Separable space">separable</a> inner product space has an orthonormal basis. </p><p>Using the <a href="/wiki/Hausdorff_maximal_principle" title="Hausdorff maximal principle">Hausdorff maximal principle</a> and the fact that in a <a href="/wiki/Hilbert_space" title="Hilbert space">complete inner product space</a> orthogonal projection onto linear subspaces is well-defined, one may also show that </p><p><b>Theorem.</b> Any <a href="/wiki/Hilbert_space" title="Hilbert space">complete inner product space</a> has an orthonormal basis. </p><p>The two previous theorems raise the question of whether all inner product spaces have an orthonormal basis. The answer, it turns out is negative. This is a non-trivial result, and is proved below. The following proof is taken from Halmos's <i>A Hilbert Space Problem Book</i> (see the references).<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (October 2017)">citation needed</span></a></i>]</sup> </p> <dl><dd><table class="toccolours collapsible collapsed" width="90%" style="text-align:left"> <tbody><tr> <th>Proof </th></tr> <tr> <td>Recall that the dimension of an inner product space is the <a href="/wiki/Cardinality" title="Cardinality">cardinality</a> of a maximal orthonormal system that it contains (by <a href="/wiki/Zorn%27s_lemma" title="Zorn's lemma">Zorn's lemma</a> it contains at least one, and any two have the same cardinality). An orthonormal basis is certainly a maximal orthonormal system but the converse need not hold in general. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> is a dense subspace of an inner product space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ace9595e3ce66fdec7e9d30202626accd676b11e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.434ex; height:2.509ex;" alt="{\displaystyle V,}"></span> then any orthonormal basis for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> is automatically an orthonormal basis for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2661a49b86bd1a5548e527bbfb068aa9f59978" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.434ex; height:2.176ex;" alt="{\displaystyle V.}"></span> Thus, it suffices to construct an inner product space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> with a dense subspace <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> whose dimension is strictly smaller than that of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2661a49b86bd1a5548e527bbfb068aa9f59978" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.434ex; height:2.176ex;" alt="{\displaystyle V.}"></span> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> be a <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> of dimension <a href="/wiki/Aleph-null" class="mw-redirect" title="Aleph-null"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{0}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e923931b35d43724d1476718cc78e813be053b07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.121ex; height:2.509ex;" alt="{\displaystyle \aleph _{0}.}"></span></a> (for instance, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\ell ^{2}(\mathbb {N} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=\ell ^{2}(\mathbb {N} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/656078a1ac7e0aecfba3df094e93dd5562a8ce03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.676ex; height:3.176ex;" alt="{\displaystyle K=\ell ^{2}(\mathbb {N} )}"></span>). Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> be an orthonormal basis of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fea4b5006afa0eb81c21b577efe304ab545d571" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle K,}"></span> so <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |E|=\aleph _{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |E|=\aleph _{0}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74cfa40daa433bf571dd8d3b2ba5173b05c96b23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.289ex; height:2.843ex;" alt="{\displaystyle |E|=\aleph _{0}.}"></span> Extend <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> to a <a href="/wiki/Basis_(linear_algebra)#Related_notions" title="Basis (linear algebra)">Hamel basis</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\cup F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>∪</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\cup F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4be5e4f6b6ada64e2b11ae3128c7eb15d2a140d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.099ex; height:2.176ex;" alt="{\displaystyle E\cup F}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fea4b5006afa0eb81c21b577efe304ab545d571" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle K,}"></span>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\cap F=\varnothing .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>∩</mo> <mi>F</mi> <mo>=</mo> <mi class="MJX-variant">∅</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\cap F=\varnothing .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c8f6086fc4d1b6af772ee1479ff5aca4cdc514a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.652ex; height:2.176ex;" alt="{\displaystyle E\cap F=\varnothing .}"></span> Since it is known that the <a href="/wiki/Hamel_dimension" class="mw-redirect" title="Hamel dimension">Hamel dimension</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae5e8f9eb465084d3a00a24026b80652b74ef58e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.654ex; height:2.009ex;" alt="{\displaystyle c,}"></span> the cardinality of the continuum, it must be that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |F|=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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |F|=c.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c775ce1630f2f1564623ab374614ddd538645e12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.787ex; height:2.843ex;" alt="{\displaystyle |F|=c.}"></span> </p><p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> be a Hilbert space of dimension <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> (for instance, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=\ell ^{2}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </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=\ell ^{2}(\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a296b641e290f87677a7f166c232c2a51ee4d521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.193ex; height:3.176ex;" alt="{\displaystyle L=\ell ^{2}(\mathbb {R} )}"></span>). Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> be an orthonormal basis for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> and let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi :F\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>:</mo> <mi>F</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi :F\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d217fe05b1e48bd7b967986a9d607b967f7d210b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.576ex; height:2.676ex;" alt="{\displaystyle \varphi :F\to B}"></span> be a bijection. Then there is a linear transformation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T:K\to L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mi>K</mi> <mo stretchy="false">→</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T:K\to L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba3d9213e37a2c614809c0a444997e9c1611e916" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.836ex; height:2.176ex;" alt="{\displaystyle T:K\to L}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Tf=\varphi (f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>f</mi> <mo>=</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Tf=\varphi (f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aed83d8ebd0a7cdbfaf96fe2f8751445a3a21ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.621ex; height:2.843ex;" alt="{\displaystyle Tf=\varphi (f)}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in F,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <mi>F</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in F,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94c70d74f4fac47c26b851cde2d74a57c94d6a25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.507ex; height:2.509ex;" alt="{\displaystyle f\in F,}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Te=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>e</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Te=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b76984792442a98f250731a9a931d254a8d46970" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.981ex; height:2.176ex;" alt="{\displaystyle Te=0}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\in E.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>∈</mo> <mi>E</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\in E.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49a310c32d8a00ffc0ef8d1bd96793f617e065e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.347ex; height:2.176ex;" alt="{\displaystyle e\in E.}"></span> </p><p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=K\oplus L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mi>K</mi> <mo>⊕</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=K\oplus L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ddd024f3c0ef626dacced6e4237337924b068be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.375ex; height:2.343ex;" alt="{\displaystyle V=K\oplus L}"></span> and let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=\{(k,Tk):k\in K\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>T</mi> <mi>k</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>k</mi> <mo>∈</mo> <mi>K</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=\{(k,Tk):k\in K\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e803ec751d6f63ed63f43be5dc8d3d9a8c3b520" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.207ex; height:2.843ex;" alt="{\displaystyle G=\{(k,Tk):k\in K\}}"></span> be the graph of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4de28b735beca1303bc5da9ba518a5a22a70a5d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.283ex; height:2.176ex;" alt="{\displaystyle T.}"></span> Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>G</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56491b3b8403461a746482cb3eb0d8e2ecbbd7d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.942ex; height:3.009ex;" alt="{\displaystyle {\overline {G}}}"></span> be the closure of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>; we will show <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {G}}=V.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>G</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {G}}=V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a7257a5883de1f10e80e709a4d116e2d4e1cacf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.474ex; height:3.009ex;" alt="{\displaystyle {\overline {G}}=V.}"></span> Since for any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\in E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>∈</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\in E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34778736d9c6d607a4da3d25594b38dd3e8c82ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.7ex; height:2.176ex;" alt="{\displaystyle e\in E}"></span> we have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (e,0)\in G,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>e</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>G</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (e,0)\in G,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8729c227a25d354bfc35f5cc24455212f71578d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.403ex; height:2.843ex;" alt="{\displaystyle (e,0)\in G,}"></span> it follows that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\oplus 0\subseteq {\overline {G}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>⊕</mo> <mn>0</mn> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>G</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\oplus 0\subseteq {\overline {G}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e712d10b58f717265da3464c35833802e5a220e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.756ex; height:3.176ex;" alt="{\displaystyle K\oplus 0\subseteq {\overline {G}}.}"></span> </p><p>Next, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in B,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>∈</mo> <mi>B</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\in B,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0c119df3e85bce0baaa3cc0df34f9d5f6eb15c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.249ex; height:2.509ex;" alt="{\displaystyle b\in B,}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=Tf}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mi>T</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=Tf}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c63b6bfbf5fe3d38802a8afcb6493e6a79227b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.011ex; height:2.509ex;" alt="{\displaystyle b=Tf}"></span> for some <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in F\subseteq K,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <mi>F</mi> <mo>⊆</mo> <mi>K</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in F\subseteq K,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84b46ed3fdce901535f31e2bb86c693bd20fe223" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.671ex; height:2.509ex;" alt="{\displaystyle f\in F\subseteq K,}"></span> so <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f,b)\in G\subseteq {\overline {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>G</mi> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>G</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f,b)\in G\subseteq {\overline {G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2fb6f0f0ef502f49eb743deae8efdad4c29d7cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.827ex; height:3.509ex;" alt="{\displaystyle (f,b)\in G\subseteq {\overline {G}}}"></span>; since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f,0)\in {\overline {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>G</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f,0)\in {\overline {G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/717bd5a5033a95e6d3479a02d39d6f1aa17f9381" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.067ex; height:3.509ex;" alt="{\displaystyle (f,0)\in {\overline {G}}}"></span> as well, we also have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,b)\in {\overline {G}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>G</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,b)\in {\overline {G}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69159e82bad1bee0f91265d61c4d0d4440b28d61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.432ex; height:3.509ex;" alt="{\displaystyle (0,b)\in {\overline {G}}.}"></span> It follows that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\oplus L\subseteq {\overline {G}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>⊕</mo> <mi>L</mi> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>G</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\oplus L\subseteq {\overline {G}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d36d31a77288f8d8c9c3ac77b71028010a60f92f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.273ex; height:3.343ex;" alt="{\displaystyle 0\oplus L\subseteq {\overline {G}},}"></span> so <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {G}}=V,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>G</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mi>V</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {G}}=V,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/574ea0bc94b9211712358b784e7302d64039ca0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.474ex; height:3.343ex;" alt="{\displaystyle {\overline {G}}=V,}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> is dense in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2661a49b86bd1a5548e527bbfb068aa9f59978" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.434ex; height:2.176ex;" alt="{\displaystyle V.}"></span> </p><p>Finally, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(e,0):e\in E\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>e</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>:</mo> <mi>e</mi> <mo>∈</mo> <mi>E</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(e,0):e\in E\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e17b1f4e61ef1231c647c082ac120100d245e39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.051ex; height:2.843ex;" alt="{\displaystyle \{(e,0):e\in E\}}"></span> is a maximal orthonormal set in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>; if <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=\langle (e,0),(k,Tk)\rangle =\langle e,k\rangle +\langle 0,Tk\rangle =\langle e,k\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mo stretchy="false">(</mo> <mi>e</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>T</mi> <mi>k</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>e</mi> <mo>,</mo> <mi>k</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mi>k</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>e</mi> <mo>,</mo> <mi>k</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=\langle (e,0),(k,Tk)\rangle =\langle e,k\rangle +\langle 0,Tk\rangle =\langle e,k\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88821a6137a5d04ad3a3ae998ebf162ad9c439f3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.262ex; height:2.843ex;" alt="{\displaystyle 0=\langle (e,0),(k,Tk)\rangle =\langle e,k\rangle +\langle 0,Tk\rangle =\langle e,k\rangle }"></span> for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\in E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>∈</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\in E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34778736d9c6d607a4da3d25594b38dd3e8c82ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.7ex; height:2.176ex;" alt="{\displaystyle e\in E}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa0825f43b484e7021f0d28d2e0d6b94091a1e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.119ex; height:2.509ex;" alt="{\displaystyle k=0,}"></span> so <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k,Tk)=(0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>T</mi> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (k,Tk)=(0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0faf6766907a8719331ebb4b09e31735dfee793f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.169ex; height:2.843ex;" alt="{\displaystyle (k,Tk)=(0,0)}"></span> is the zero vector in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc645a5b7e8a2022ad70fc42dbda04c008a33a9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.474ex; height:2.176ex;" alt="{\displaystyle G.}"></span> Hence the dimension of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |E|=\aleph _{0},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |E|=\aleph _{0},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9bd997bcc84934fbd82c6c4554d7d39bbafae93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.289ex; height:2.843ex;" alt="{\displaystyle |E|=\aleph _{0},}"></span> whereas it is clear that the dimension of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13b8d90daa52ffa8e5988459b6f10ef4d64ee5da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.654ex; height:1.676ex;" alt="{\displaystyle c.}"></span> This completes the proof. </p> </td></tr></tbody></table></dd></dl> <p><a href="/wiki/Parseval%27s_identity" title="Parseval's identity">Parseval's identity</a> leads immediately to the following theorem: </p><p><b>Theorem.</b> Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> be a separable inner product space and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{e_{k}\right\}_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>{</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{e_{k}\right\}_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/277380fd3ae81a149d77925de57993736e1a81fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.586ex; height:3.009ex;" alt="{\displaystyle \left\{e_{k}\right\}_{k}}"></span> an orthonormal basis of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2661a49b86bd1a5548e527bbfb068aa9f59978" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.434ex; height:2.176ex;" alt="{\displaystyle V.}"></span> Then the map <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto {\bigl \{}\langle e_{k},x\rangle {\bigr \}}_{k\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <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> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto {\bigl \{}\langle e_{k},x\rangle {\bigr \}}_{k\in \mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b38f38f737c73a460876f0e3b8eed48a34650364" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.371ex; height:3.343ex;" alt="{\displaystyle x\mapsto {\bigl \{}\langle e_{k},x\rangle {\bigr \}}_{k\in \mathbb {N} }}"></span> is an isometric linear map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\rightarrow \ell ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">→</mo> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\rightarrow \ell ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8071f271637b5f4f89b7c3b51ff29cdc5bc7e3d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.425ex; height:2.676ex;" alt="{\displaystyle V\rightarrow \ell ^{2}}"></span> with a dense image. </p><p>This theorem can be regarded as an abstract form of <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a>, in which an arbitrary orthonormal basis plays the role of the sequence of <a href="/wiki/Trigonometric_polynomial" title="Trigonometric polynomial">trigonometric polynomials</a>. Note that the underlying index set can be taken to be any countable set (and in fact any set whatsoever, provided <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}}"></span> is defined appropriately, as is explained in the article <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a>). In particular, we obtain the following result in the theory of Fourier series: </p><p><b>Theorem.</b> Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> be the inner product space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C[-\pi ,\pi ].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">[</mo> <mo>−</mo> <mi>π</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C[-\pi ,\pi ].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/223f9d7493adedbd3d9cf0c35f812f8236c202d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.213ex; height:2.843ex;" alt="{\displaystyle C[-\pi ,\pi ].}"></span> Then the sequence (indexed on set of all integers) of continuous functions <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{k}(t)={\frac {e^{ikt}}{\sqrt {2\pi }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>t</mi> </mrow> </msup> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{k}(t)={\frac {e^{ikt}}{\sqrt {2\pi }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67231686f14dfea4f61da5fd81d1e3425020bcef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:13.186ex; height:6.676ex;" alt="{\displaystyle e_{k}(t)={\frac {e^{ikt}}{\sqrt {2\pi }}}}"></span> is an orthonormal basis of the space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C[-\pi ,\pi ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">[</mo> <mo>−</mo> <mi>π</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C[-\pi ,\pi ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7004d3934596427928c52976c6c0457dcd4e522b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.566ex; height:2.843ex;" alt="{\displaystyle C[-\pi ,\pi ]}"></span> with the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}}"></span> inner product. The mapping <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\mapsto {\frac {1}{\sqrt {2\pi }}}\left\{\int _{-\pi }^{\pi }f(t)e^{-ikt}\,\mathrm {d} t\right\}_{k\in \mathbb {Z} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <msub> <mrow> <mo>{</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>k</mi> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</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\mapsto {\frac {1}{\sqrt {2\pi }}}\left\{\int _{-\pi }^{\pi }f(t)e^{-ikt}\,\mathrm {d} t\right\}_{k\in \mathbb {Z} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3047892082fa23c14a5ca6b248eb640f045c90a7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:32.117ex; height:6.509ex;" alt="{\displaystyle f\mapsto {\frac {1}{\sqrt {2\pi }}}\left\{\int _{-\pi }^{\pi }f(t)e^{-ikt}\,\mathrm {d} t\right\}_{k\in \mathbb {Z} }}"></span> is an isometric linear map with dense image. </p><p>Orthogonality of the sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{e_{k}\}_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{e_{k}\}_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/465501a55f0530a201631136abb1ef4dbce59199" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.586ex; height:2.843ex;" alt="{\displaystyle \{e_{k}\}_{k}}"></span> follows immediately from the fact that if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\neq j,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≠</mo> <mi>j</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\neq j,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7b76832728b5272b70dc1a95c3931ac8c89fbc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.915ex; height:2.676ex;" alt="{\displaystyle k\neq j,}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\pi }^{\pi }e^{-i(j-k)t}\,\mathrm {d} t=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\pi }^{\pi }e^{-i(j-k)t}\,\mathrm {d} t=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba695334ce8205b102c4055a7dbb4d659359dd8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.407ex; height:6.009ex;" alt="{\displaystyle \int _{-\pi }^{\pi }e^{-i(j-k)t}\,\mathrm {d} t=0.}"></span> </p><p>Normality of the sequence is by design, that is, the coefficients are so chosen so that the norm comes out to 1. Finally the fact that the sequence has a dense algebraic span, in the <em>inner product norm</em>, follows from the fact that the sequence has a dense algebraic span, this time in the space of continuous periodic functions on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-\pi ,\pi ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>−</mo> <mi>π</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [-\pi ,\pi ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb064fd6c55820cfa660eabeeda0f6e3c4935ae6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.8ex; height:2.843ex;" alt="{\displaystyle [-\pi ,\pi ]}"></span> with the uniform norm. This is the content of the <a href="/wiki/Weierstrass_approximation_theorem" class="mw-redirect" title="Weierstrass approximation theorem">Weierstrass theorem</a> on the uniform density of trigonometric polynomials. </p> <div class="mw-heading mw-heading2"><h2 id="Operators_on_inner_product_spaces">Operators on inner product spaces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=18" title="Edit section: Operators on inner product spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Operator_theory" title="Operator theory">Operator theory</a></div> <p>Several types of <a href="/wiki/Linear" class="mw-redirect" title="Linear">linear</a> maps <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A:V\to W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A:V\to W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97b4b6380b2f4700cea977536a2af33d51efb89f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.517ex; height:2.176ex;" alt="{\displaystyle A:V\to W}"></span> between inner product spaces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> are of relevance: </p> <ul><li><em><a href="/wiki/Continuous_linear_operator" title="Continuous linear operator">Continuous linear maps</a></em>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A:V\to W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A:V\to W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97b4b6380b2f4700cea977536a2af33d51efb89f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.517ex; height:2.176ex;" alt="{\displaystyle A:V\to W}"></span> is linear and continuous with respect to the metric defined above, or equivalently, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is linear and the set of non-negative reals <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\|Ax\|:\|x\|\leq 1\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">‖</mo> <mi>A</mi> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>:</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\|Ax\|:\|x\|\leq 1\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a07632d041f42c21bd2b19e6712e0dabf076593e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.222ex; height:2.843ex;" alt="{\displaystyle \{\|Ax\|:\|x\|\leq 1\},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> ranges over the closed unit ball of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ace9595e3ce66fdec7e9d30202626accd676b11e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.434ex; height:2.509ex;" alt="{\displaystyle V,}"></span> is bounded.</li> <li><em>Symmetric linear operators</em>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A:V\to W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A:V\to W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97b4b6380b2f4700cea977536a2af33d51efb89f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.517ex; height:2.176ex;" alt="{\displaystyle A:V\to W}"></span> is linear and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>A</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>A</mi> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b148dd9277ab2b18762deba9d034b58832bb025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.241ex; height:2.843ex;" alt="{\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle }"></span> for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in V.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afae3d54ba364e742c2efa040fc447300966c7f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.794ex; height:2.509ex;" alt="{\displaystyle x,y\in V.}"></span></li> <li><em><a href="/wiki/Isometry" title="Isometry">Isometries</a></em>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A:V\to W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A:V\to W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97b4b6380b2f4700cea977536a2af33d51efb89f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.517ex; height:2.176ex;" alt="{\displaystyle A:V\to W}"></span> satisfies <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|Ax\|=\|x\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>A</mi> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|Ax\|=\|x\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec1a9ebb4090d411c9dc8918686dba5f0074a68f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.151ex; height:2.843ex;" alt="{\displaystyle \|Ax\|=\|x\|}"></span> for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in V.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2db8a2c9b98774154e833074ffbab1d3fc6c0f61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.604ex; height:2.176ex;" alt="{\displaystyle x\in V.}"></span> A <em>linear isometry</em> (resp. an <em><a href="/wiki/Antilinear_map" title="Antilinear map">antilinear</a> isometry</em>) is an isometry that is also a linear map (resp. an <a href="/wiki/Antilinear_map" title="Antilinear map">antilinear map</a>). For inner product spaces, the <a href="/wiki/Polarization_identity" title="Polarization identity">polarization identity</a> can be used to show that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is an isometry if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle Ax,Ay\rangle =\langle x,y\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>A</mi> <mi>x</mi> <mo>,</mo> <mi>A</mi> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle Ax,Ay\rangle =\langle x,y\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01bf4848daedaecda5cc5775bb8532a07b1a244e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.241ex; height:2.843ex;" alt="{\displaystyle \langle Ax,Ay\rangle =\langle x,y\rangle }"></span> for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in V.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afae3d54ba364e742c2efa040fc447300966c7f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.794ex; height:2.509ex;" alt="{\displaystyle x,y\in V.}"></span> All isometries are <a href="/wiki/Injective" class="mw-redirect" title="Injective">injective</a>. The <a href="/wiki/Mazur%E2%80%93Ulam_theorem" title="Mazur–Ulam theorem">Mazur–Ulam theorem</a> establishes that every surjective isometry between two <em>real</em> normed spaces is an <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformation</a>. Consequently, an isometry <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> between real inner product spaces is a linear map if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(0)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</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 A(0)=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/723c250806517cd94ac42c8707be95934bca01a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.623ex; height:2.843ex;" alt="{\displaystyle A(0)=0.}"></span> Isometries are <a href="/wiki/Morphism" title="Morphism">morphisms</a> between inner product spaces, and morphisms of real inner product spaces are orthogonal transformations (compare with <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal matrix</a>).</li> <li><em>Isometrical isomorphisms</em>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A:V\to W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A:V\to W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97b4b6380b2f4700cea977536a2af33d51efb89f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.517ex; height:2.176ex;" alt="{\displaystyle A:V\to W}"></span> is an isometry which is <a href="/wiki/Surjective" class="mw-redirect" title="Surjective">surjective</a> (and hence <a href="/wiki/Bijective" class="mw-redirect" title="Bijective">bijective</a>). Isometrical isomorphisms are also known as unitary operators (compare with <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary matrix</a>).</li></ul> <p>From the point of view of inner product space theory, there is no need to distinguish between two spaces which are isometrically isomorphic. The <a href="/wiki/Spectral_theorem" title="Spectral theorem">spectral theorem</a> provides a canonical form for symmetric, unitary and more generally <a href="/wiki/Normal_operator" title="Normal operator">normal operators</a> on finite dimensional inner product spaces. A generalization of the spectral theorem holds for continuous normal operators in Hilbert spaces.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=19" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any of the axioms of an inner product may be weakened, yielding generalized notions. The generalizations that are closest to inner products occur where bilinearity and conjugate symmetry are retained, but positive-definiteness is weakened. </p> <div class="mw-heading mw-heading3"><h3 id="Degenerate_inner_products">Degenerate inner products</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=20" title="Edit section: Degenerate inner products"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Krein_space" class="mw-redirect" title="Krein space">Krein space</a></div> <p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> is a vector space and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="thinmathspace" /> <mo>⋅</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \,\cdot \,,\,\cdot \,\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2027b7f5f03c87f88b943b87e1062cc837e60dd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.685ex; height:2.843ex;" alt="{\displaystyle \langle \,\cdot \,,\,\cdot \,\rangle }"></span> a semi-definite sesquilinear form, then the function: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|={\sqrt {\langle x,x\rangle }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|={\sqrt {\langle x,x\rangle }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a885f02102ba1179708035546be39be936979b3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.579ex; height:4.843ex;" alt="{\displaystyle \|x\|={\sqrt {\langle x,x\rangle }}}"></span> makes sense and satisfies all the properties of norm except that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|x\|=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/787cc4ec9f7efa527a924e4104b6e6501a949f06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.916ex; height:2.843ex;" alt="{\displaystyle \|x\|=0}"></span> does not imply <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"></span> (such a functional is then called a <a href="/wiki/Semi-norm" class="mw-redirect" title="Semi-norm">semi-norm</a>). We can produce an inner product space by considering the quotient <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=V/\{x:\|x\|=0\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>:</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=V/\{x:\|x\|=0\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85743d99fceef5471b0ebc2c7ac7ff151594b9ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.638ex; height:2.843ex;" alt="{\displaystyle W=V/\{x:\|x\|=0\}.}"></span> The sesquilinear form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="thinmathspace" /> <mo>⋅</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \,\cdot \,,\,\cdot \,\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2027b7f5f03c87f88b943b87e1062cc837e60dd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.685ex; height:2.843ex;" alt="{\displaystyle \langle \,\cdot \,,\,\cdot \,\rangle }"></span> factors through <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/035a080f11445ba1e7745704a6031989a311a7d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.082ex; height:2.176ex;" alt="{\displaystyle W.}"></span> </p><p>This construction is used in numerous contexts. The <a href="/wiki/Gelfand%E2%80%93Naimark%E2%80%93Segal_construction" title="Gelfand–Naimark–Segal construction">Gelfand–Naimark–Segal construction</a> is a particularly important example of the use of this technique. Another example is the representation of <a href="/wiki/Mercer%27s_theorem" title="Mercer's theorem">semi-definite kernels</a> on arbitrary sets. </p> <div class="mw-heading mw-heading3"><h3 id="Nondegenerate_conjugate_symmetric_forms">Nondegenerate conjugate symmetric forms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=21" title="Edit section: Nondegenerate conjugate symmetric forms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Pseudo-Euclidean_space" title="Pseudo-Euclidean space">Pseudo-Euclidean space</a></div> <p>Alternatively, one may require that the pairing be a <a href="/wiki/Nondegenerate_form" class="mw-redirect" title="Nondegenerate form">nondegenerate form</a>, meaning that for all non-zero <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35a455db7b2aab1b0e72ccbc7385e4424e2372e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.676ex;" alt="{\displaystyle x\neq 0}"></span> there exists some <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle \neq 0,}"> <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> <mo fence="false" stretchy="false">⟩</mo> <mo>≠</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle \neq 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70cad7232a67fb3c44d692c1614cfe977e991705" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.236ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle \neq 0,}"></span> though <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> need not equal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>; in other words, the induced map to the dual space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\to V^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">→</mo> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\to V^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e86ea24eb05dd60cbcf6eb787a5f07e3861984e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.372ex; height:2.343ex;" alt="{\displaystyle V\to V^{*}}"></span> is injective. This generalization is important in <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>: a manifold whose tangent spaces have an inner product is a <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>, while if this is related to nondegenerate conjugate symmetric form the manifold is a <a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">pseudo-Riemannian manifold</a>. By <a href="/wiki/Sylvester%27s_law_of_inertia" title="Sylvester's law of inertia">Sylvester's law of inertia</a>, just as every inner product is similar to the dot product with positive weights on a set of vectors, every nondegenerate conjugate symmetric form is similar to the dot product with <em>nonzero</em> weights on a set of vectors, and the number of positive and negative weights are called respectively the positive index and negative index. Product of vectors in <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a> is an example of indefinite inner product, although, technically speaking, it is not an inner product according to the standard definition above. Minkowski space has four <a href="/wiki/Dimension_(mathematics)" class="mw-redirect" title="Dimension (mathematics)">dimensions</a> and indices 3 and 1 (assignment of <a href="/wiki/Sign_(mathematics)" title="Sign (mathematics)">"+" and "−"</a> to them <a href="/wiki/Sign_convention#Metric_signature" title="Sign convention">differs depending on conventions</a>). </p><p>Purely algebraic statements (ones that do not use positivity) usually only rely on the nondegeneracy (the injective homomorphism <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\to V^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">→</mo> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\to V^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e86ea24eb05dd60cbcf6eb787a5f07e3861984e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.372ex; height:2.343ex;" alt="{\displaystyle V\to V^{*}}"></span>) and thus hold more generally. </p> <div class="mw-heading mw-heading2"><h2 id="Related_products">Related products</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=22" title="Edit section: Related products"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The term "inner product" is opposed to <a href="/wiki/Outer_product" title="Outer product">outer product</a> (<a href="/wiki/Tensor_product" title="Tensor product">tensor product</a>), which is a slightly more general opposite. Simply, in coordinates, the inner product is the product of a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bce5f6a6d0d32834484048c16f3b39f9c23d076" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.398ex; height:2.176ex;" alt="{\displaystyle 1\times n}"></span> <em>covector</em> with an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d24148f103e1cccb60addeeb0a64cb1c3d5622e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.398ex; height:2.176ex;" alt="{\displaystyle n\times 1}"></span> vector, yielding a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\times 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>×</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\times 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b4bf91a527dc01af9ef6ace81199becf1308e00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 1\times 1}"></span> matrix (a scalar), while the outer product is the product of an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f7ea91d81567531f5ef6d3b669be211ff953e6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.043ex; height:2.176ex;" alt="{\displaystyle m\times 1}"></span> vector with a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bce5f6a6d0d32834484048c16f3b39f9c23d076" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.398ex; height:2.176ex;" alt="{\displaystyle 1\times n}"></span> covector, yielding an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span> matrix. The outer product is defined for different dimensions, while the inner product requires the same dimension. If the dimensions are the same, then the inner product is the <em><a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a></em> of the outer product (trace only being properly defined for square matrices). In an informal summary: "inner is horizontal times vertical and shrinks down, outer is vertical times horizontal and expands out". </p><p>More abstractly, the outer product is the bilinear map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W\times V^{*}\to \hom(V,W)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>×</mo> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">→</mo> <mi>hom</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mi>W</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W\times V^{*}\to \hom(V,W)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/230a81e4f53292e6c62c20013a939a0358bf23eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.317ex; height:2.843ex;" alt="{\displaystyle W\times V^{*}\to \hom(V,W)}"></span> sending a vector and a covector to a rank 1 linear transformation (<a href="/wiki/Simple_tensor" class="mw-redirect" title="Simple tensor">simple tensor</a> of type (1, 1)), while the inner product is the bilinear evaluation map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{*}\times V\to F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>×</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V^{*}\times V\to F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9e94fbfe0ac2b30ba95a9d76c33166b3b4323ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.954ex; height:2.343ex;" alt="{\displaystyle V^{*}\times V\to F}"></span> given by evaluating a covector on a vector; the order of the domain vector spaces here reflects the covector/vector distinction. </p><p>The inner product and outer product should not be confused with the <a href="/wiki/Interior_product" title="Interior product">interior product</a> and <a href="/wiki/Exterior_product" class="mw-redirect" title="Exterior product">exterior product</a>, which are instead operations on <a href="/wiki/Vector_field" title="Vector field">vector fields</a> and <a href="/wiki/Differential_form" title="Differential form">differential forms</a>, or more generally on the <a href="/wiki/Exterior_algebra" title="Exterior algebra">exterior algebra</a>. </p><p>As a further complication, in <a href="/wiki/Geometric_algebra" title="Geometric algebra">geometric algebra</a> the inner product and the <em>exterior</em> (Grassmann) product are combined in the geometric product (the Clifford product in a <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a>) – the inner product sends two vectors (1-vectors) to a scalar (a 0-vector), while the exterior product sends two vectors to a bivector (2-vector) – and in this context the exterior product is usually called the <em>outer product</em> (alternatively, <em><a href="/wiki/Wedge_product" class="mw-redirect" title="Wedge product">wedge product</a></em>). The inner product is more correctly called a <em>scalar</em> product in this context, as the nondegenerate quadratic form in question need not be positive definite (need not be an inner product). </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=23" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Bilinear_form" title="Bilinear form">Bilinear form</a> – Scalar-valued bilinear function</li> <li><a href="/wiki/Biorthogonal_system" title="Biorthogonal system">Biorthogonal system</a></li> <li><a href="/wiki/Dual_space" title="Dual space">Dual space</a> – In mathematics, vector space of linear forms</li> <li><a href="/wiki/Energetic_space" title="Energetic space">Energetic space</a> – subspace of a given real Hilbert space equipped with a new "energetic" inner product<span style="display:none" class="category-wikidata-fallback-annotation">Pages displaying wikidata descriptions as a fallback</span></li> <li><a href="/wiki/L-semi-inner_product" title="L-semi-inner product">L-semi-inner product</a> – Generalization of inner products that applies to all normed spaces</li> <li><a href="/wiki/Minkowski_distance" title="Minkowski distance">Minkowski distance</a> – Mathematical metric in normed vector space</li> <li><a href="/wiki/Orthogonal_basis" title="Orthogonal basis">Orthogonal basis</a> – Basis for v whose vectors are mutually orthogonal</li> <li><a href="/wiki/Orthogonal_complement" title="Orthogonal complement">Orthogonal complement</a> – Concept in linear algebra</li> <li><a href="/wiki/Orthonormal_basis" title="Orthonormal basis">Orthonormal basis</a> – Specific linear basis (mathematics)</li> <li><a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=24" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"><ol class="references"> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">By combining the <i>linear in the first argument</i> property with the <i>conjugate symmetry</i> property you get <i>conjugate-linear in the second argument</i>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \langle x,by\rangle =\langle x,y\rangle {\overline {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>b</mi> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \langle x,by\rangle =\langle x,y\rangle {\overline {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b0714b329562d285ff925bd14e7ad7c63412a94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.865ex; height:3.509ex;" alt="{\textstyle \langle x,by\rangle =\langle x,y\rangle {\overline {b}}}"></span>. This is how the inner product was originally defined and is used in most mathematical contexts. A different convention has been adopted in theoretical physics and quantum mechanics, originating in the <a href="/wiki/Bra-ket" class="mw-redirect" title="Bra-ket">bra-ket</a> notation of <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a>, where the inner product is taken to be <i>linear in the second argument</i> and <i>conjugate-linear in the first argument</i>; this convention is used in many other domains such as engineering and computer science.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=25" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-FOOTNOTETrèves2006112–125-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTETrèves2006112–125_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTETrèves2006112–125_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTETrèves2006112–125_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFTrèves2006">Trèves 2006</a>, pp. 112–125.</span> </li> <li id="cite_note-FOOTNOTESchaeferWolff199940–45-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESchaeferWolff199940–45_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSchaeferWolff1999">Schaefer & Wolff 1999</a>, pp. 40–45.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><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="CITEREFMoore1995" class="citation journal cs1">Moore, Gregory H. (1995). <a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fhmat.1995.1025">"The axiomatization of linear algebra: 1875-1940"</a>. <i>Historia Mathematica</i>. <b>22</b> (3): <span class="nowrap">262–</span>303. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fhmat.1995.1025">10.1006/hmat.1995.1025</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Historia+Mathematica&rft.atitle=The+axiomatization+of+linear+algebra%3A+1875-1940&rft.volume=22&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E262-%3C%2Fspan%3E303&rft.date=1995&rft_id=info%3Adoi%2F10.1006%2Fhmat.1995.1025&rft.aulast=Moore&rft.aufirst=Gregory+H.&rft_id=https%3A%2F%2Fdoi.org%2F10.1006%252Fhmat.1995.1025&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInner+product+space" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTESchaeferWolff199936–72-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESchaeferWolff199936–72_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSchaeferWolff1999">Schaefer & Wolff 1999</a>, pp. 36–72.</span> </li> <li id="cite_note-Jain-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Jain_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJainAhmad1995" class="citation book cs1">Jain, P. 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(1967). <a rel="nofollow" class="external text" href="https://www.tandfonline.com/doi/pdf/10.1080/0025570X.1967.11975804">"Ptolemy's Inequality and the Chordal Metric"</a>. <i>Mathematics Magazine</i>. <b>40</b> (5): <span class="nowrap">233–</span>235. <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%2F2688275">10.2307/2688275</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/2688275">2688275</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=Ptolemy%27s+Inequality+and+the+Chordal+Metric&rft.volume=40&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E233-%3C%2Fspan%3E235&rft.date=1967&rft_id=info%3Adoi%2F10.2307%2F2688275&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2688275%23id-name%3DJSTOR&rft.aulast=Apostol&rft.aufirst=Tom+M.&rft_id=https%3A%2F%2Fwww.tandfonline.com%2Fdoi%2Fpdf%2F10.1080%2F0025570X.1967.11975804&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInner+product+space" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTERudin1991306–312-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTERudin1991306–312_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTERudin1991306–312_13-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFRudin1991">Rudin 1991</a>, pp. 306–312.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a href="#CITEREFRudin1991">Rudin 1991</a></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inner_product_space&action=edit&section=26" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAxler1997" class="citation book cs1">Axler, Sheldon (1997). <i>Linear Algebra Done Right</i> (2nd ed.). Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-98258-8" title="Special:BookSources/978-0-387-98258-8"><bdi>978-0-387-98258-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+Done+Right&rft.place=Berlin%2C+New+York&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1997&rft.isbn=978-0-387-98258-8&rft.aulast=Axler&rft.aufirst=Sheldon&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInner+product+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDieudonné1969" class="citation book cs1"><a href="/wiki/Jean_Dieudonn%C3%A9" title="Jean Dieudonné">Dieudonné, Jean</a> (1969). <i>Treatise on Analysis, Vol. I [Foundations of Modern Analysis]</i> (2nd ed.). <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4067-2791-3" title="Special:BookSources/978-1-4067-2791-3"><bdi>978-1-4067-2791-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Treatise+on+Analysis%2C+Vol.+I+%5BFoundations+of+Modern+Analysis%5D&rft.edition=2nd&rft.pub=Academic+Press&rft.date=1969&rft.isbn=978-1-4067-2791-3&rft.aulast=Dieudonn%C3%A9&rft.aufirst=Jean&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInner+product+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEmch1972" class="citation book cs1">Emch, Gerard G. (1972). <i>Algebraic Methods in Statistical Mechanics and Quantum Field Theory</i>. <a href="/wiki/Wiley-Interscience" class="mw-redirect" title="Wiley-Interscience">Wiley-Interscience</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-23900-0" title="Special:BookSources/978-0-471-23900-0"><bdi>978-0-471-23900-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Methods+in+Statistical+Mechanics+and+Quantum+Field+Theory&rft.pub=Wiley-Interscience&rft.date=1972&rft.isbn=978-0-471-23900-0&rft.aulast=Emch&rft.aufirst=Gerard+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInner+product+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalmos1982" class="citation book cs1"><a href="/wiki/Paul_Richard_Halmos" class="mw-redirect" title="Paul Richard Halmos">Halmos, Paul R.</a> (8 November 1982). <i>A Hilbert Space Problem Book</i>. <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>. 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New York: <a href="/wiki/Springer_Publishing" title="Springer Publishing">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90685-0" title="Special:BookSources/978-0-387-90685-0"><bdi>978-0-387-90685-0</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/8169781">8169781</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Hilbert+Space+Problem+Book&rft.place=New+York&rft.series=Graduate+Texts+in+Mathematics&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1982-11-08&rft_id=info%3Aoclcnum%2F8169781&rft.isbn=978-0-387-90685-0&rft.aulast=Halmos&rft.aufirst=Paul+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInner+product+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLax2002" class="citation book cs1"><a href="/wiki/Peter_D._Lax" class="mw-redirect" title="Peter D. Lax">Lax, Peter D.</a> (2002). <a rel="nofollow" class="external text" href="http://www.math.univ-metz.fr/~gnc/bibliographie/Functional%20Analysis/Lax,.Functional.Analysis,.Wiley,.2002,.603s.pdf"><i>Functional Analysis</i></a> <span class="cs1-format">(PDF)</span>. Pure and Applied Mathematics. New York: Wiley-Interscience. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-55604-6" title="Special:BookSources/978-0-471-55604-6"><bdi>978-0-471-55604-6</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/47767143">47767143</a><span class="reference-accessdate">. Retrieved <span class="nowrap">July 22,</span> 2020</span>.</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&rft.series=Pure+and+Applied+Mathematics&rft.pub=Wiley-Interscience&rft.date=2002&rft_id=info%3Aoclcnum%2F47767143&rft.isbn=978-0-471-55604-6&rft.aulast=Lax&rft.aufirst=Peter+D.&rft_id=http%3A%2F%2Fwww.math.univ-metz.fr%2F~gnc%2Fbibliographie%2FFunctional%2520Analysis%2FLax%2C.Functional.Analysis%2C.Wiley%2C.2002%2C.603s.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInner+product+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudin1991" class="citation book cs1"><a href="/wiki/Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1991). <a rel="nofollow" class="external text" href="https://archive.org/details/functionalanalys00rudi"><i>Functional Analysis</i></a>. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). 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%3AInner+product+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchaeferWolff1999" class="citation book cs1"><a href="/wiki/Helmut_H._Schaefer" title="Helmut H. 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New York, NY: Springer New York Imprint Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4612-7155-0" title="Special:BookSources/978-1-4612-7155-0"><bdi>978-1-4612-7155-0</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/840278135">840278135</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=New+York%2C+NY&rft.series=GTM&rft.edition=Second&rft.pub=Springer+New+York+Imprint+Springer&rft.date=1999&rft_id=info%3Aoclcnum%2F840278135&rft.isbn=978-1-4612-7155-0&rft.aulast=Schaefer&rft.aufirst=Helmut+H.&rft.au=Wolff%2C+Manfred+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInner+product+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchechter1996" class="citation book cs1"><a href="/wiki/Eric_Schechter" title="Eric Schechter">Schechter, Eric</a> (1996). <i>Handbook of Analysis and Its Foundations</i>. San Diego, CA: Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-622760-4" title="Special:BookSources/978-0-12-622760-4"><bdi>978-0-12-622760-4</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/175294365">175294365</a>.</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=San+Diego%2C+CA&rft.pub=Academic+Press&rft.date=1996&rft_id=info%3Aoclcnum%2F175294365&rft.isbn=978-0-12-622760-4&rft.aulast=Schechter&rft.aufirst=Eric&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInner+product+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSwartz1992" class="citation book cs1">Swartz, Charles (1992). <i>An introduction to Functional Analysis</i>. 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Dekker. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8247-8643-4" title="Special:BookSources/978-0-8247-8643-4"><bdi>978-0-8247-8643-4</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/24909067">24909067</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+Functional+Analysis&rft.place=New+York&rft.pub=M.+Dekker&rft.date=1992&rft_id=info%3Aoclcnum%2F24909067&rft.isbn=978-0-8247-8643-4&rft.aulast=Swartz&rft.aufirst=Charles&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInner+product+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTrèves2006" class="citation book cs1"><a href="/wiki/Fran%C3%A7ois_Tr%C3%A8ves" title="François Trèves">Trèves, François</a> (2006) [1967]. <i>Topological Vector Spaces, Distributions and Kernels</i>. Mineola, N.Y.: Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-45352-1" title="Special:BookSources/978-0-486-45352-1"><bdi>978-0-486-45352-1</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/853623322">853623322</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%2C+Distributions+and+Kernels&rft.place=Mineola%2C+N.Y.&rft.pub=Dover+Publications&rft.date=2006&rft_id=info%3Aoclcnum%2F853623322&rft.isbn=978-0-486-45352-1&rft.aulast=Tr%C3%A8ves&rft.aufirst=Fran%C3%A7ois&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInner+product+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYoung1988" class="citation book cs1">Young, Nicholas (1988). <i>An Introduction to Hilbert Space</i>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-33717-5" title="Special:BookSources/978-0-521-33717-5"><bdi>978-0-521-33717-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Hilbert+Space&rft.pub=Cambridge+University+Press&rft.date=1988&rft.isbn=978-0-521-33717-5&rft.aulast=Young&rft.aufirst=Nicholas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInner+product+space" class="Z3988"></span></li> <li>Zamani, A.; Moslehian, M.S.; & Frank, M. (2015) "Angle Preserving Mappings", <i>Journal of Analysis and Applications</i> 34: 485 to 500 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4171%2FZAA%2F1551">10.4171/ZAA/1551</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol 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space">Vector space</a></li> <li><a href="/wiki/Scalar_multiplication" title="Scalar multiplication">Scalar multiplication</a></li> <li><a href="/wiki/Vector_projection" title="Vector projection">Vector projection</a></li> <li><a href="/wiki/Linear_span" title="Linear span">Linear span</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a></li> <li><a href="/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">Linear projection</a></li> <li><a href="/wiki/Linear_independence" title="Linear independence">Linear independence</a></li> <li><a href="/wiki/Linear_combination" title="Linear combination">Linear combination</a></li> <li><a href="/wiki/Multilinear_map" title="Multilinear map">Multilinear map</a></li> <li><a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li> <li><a href="/wiki/Change_of_basis" title="Change of basis">Change of basis</a></li> <li><a href="/wiki/Row_and_column_vectors" title="Row and column vectors">Row and column vectors</a></li> <li><a href="/wiki/Row_and_column_spaces" title="Row and column spaces">Row and column spaces</a></li> <li><a href="/wiki/Kernel_(linear_algebra)" title="Kernel (linear algebra)">Kernel</a></li> <li><a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">Eigenvalues and eigenvectors</a></li> <li><a href="/wiki/Transpose" title="Transpose">Transpose</a></li> <li><a href="/wiki/System_of_linear_equations" title="System of linear equations">Linear equations</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/Euclidean_space" title="Euclidean space"><img alt="Three dimensional Euclidean space" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/80px-Linear_subspaces_with_shading.svg.png" decoding="async" width="80" height="58" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/120px-Linear_subspaces_with_shading.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/160px-Linear_subspaces_with_shading.svg.png 2x" data-file-width="325" data-file-height="236" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrices</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/Block_matrix" title="Block matrix">Block</a></li> <li><a href="/wiki/Matrix_decomposition" title="Matrix decomposition">Decomposition</a></li> <li><a href="/wiki/Invertible_matrix" title="Invertible matrix">Invertible</a></li> <li><a href="/wiki/Minor_(linear_algebra)" title="Minor (linear algebra)">Minor</a></li> <li><a href="/wiki/Matrix_multiplication" title="Matrix multiplication">Multiplication</a></li> <li><a href="/wiki/Rank_(linear_algebra)" title="Rank (linear algebra)">Rank</a></li> <li><a href="/wiki/Transformation_matrix" title="Transformation matrix">Transformation</a></li> <li><a href="/wiki/Cramer%27s_rule" title="Cramer's rule">Cramer's rule</a></li> <li><a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a></li> <li><a href="/wiki/Productive_matrix" title="Productive matrix">Productive matrix</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Bilinear_map" title="Bilinear map">Bilinear</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/Orthogonality" title="Orthogonality">Orthogonality</a></li> <li><a href="/wiki/Dot_product" title="Dot product">Dot product</a></li> <li><a href="/wiki/Hadamard_product_(matrices)" title="Hadamard product (matrices)">Hadamard product</a></li> <li><a class="mw-selflink selflink">Inner product space</a></li> <li><a href="/wiki/Outer_product" title="Outer product">Outer product</a></li> <li><a href="/wiki/Kronecker_product" title="Kronecker product">Kronecker product</a></li> <li><a href="/wiki/Gram%E2%80%93Schmidt_process" title="Gram–Schmidt process">Gram–Schmidt process</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</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/Determinant" title="Determinant">Determinant</a></li> <li><a href="/wiki/Cross_product" title="Cross product">Cross product</a></li> <li><a href="/wiki/Triple_product" title="Triple product">Triple product</a></li> <li><a href="/wiki/Seven-dimensional_cross_product" title="Seven-dimensional cross product">Seven-dimensional cross product</a></li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a></li> <li><a href="/wiki/Exterior_algebra" title="Exterior algebra">Exterior algebra</a></li> <li><a href="/wiki/Bivector" title="Bivector">Bivector</a></li> <li><a href="/wiki/Multivector" title="Multivector">Multivector</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a></li> <li><a href="/wiki/Outermorphism" title="Outermorphism">Outermorphism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Vector_space" title="Vector space">Vector space</a> constructions</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/Dual_space" title="Dual space">Dual</a></li> <li><a href="/wiki/Direct_sum_of_modules#Construction_for_two_vector_spaces" title="Direct sum of modules">Direct sum</a></li> <li><a href="/wiki/Function_space#In_linear_algebra" title="Function space">Function space</a></li> <li><a href="/wiki/Quotient_space_(linear_algebra)" title="Quotient space (linear algebra)">Quotient</a></li> <li><a href="/wiki/Linear_subspace" title="Linear subspace">Subspace</a></li> <li><a href="/wiki/Tensor_product" title="Tensor product">Tensor product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Numerical_linear_algebra" title="Numerical linear algebra">Numerical</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/Floating-point_arithmetic" title="Floating-point arithmetic">Floating-point</a></li> <li><a href="/wiki/Numerical_stability" title="Numerical stability">Numerical stability</a></li> <li><a href="/wiki/Basic_Linear_Algebra_Subprograms" title="Basic Linear Algebra Subprograms">Basic Linear Algebra Subprograms</a></li> <li><a href="/wiki/Sparse_matrix" title="Sparse matrix">Sparse matrix</a></li> <li><a href="/wiki/Comparison_of_linear_algebra_libraries" title="Comparison of linear algebra libraries">Comparison of linear algebra libraries</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Linear_algebra" title="Category:Linear algebra">Category</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 template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Functional_analysis" title="Template talk:Functional analysis"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Functional_analysis" title="Special:EditPage/Template:Functional analysis"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Functional_analysis_(topics_–_glossary)364" style="font-size:114%;margin:0 4em"><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a> (<a href="/wiki/List_of_functional_analysis_topics" title="List of functional analysis topics">topics</a> – <a href="/wiki/Glossary_of_functional_analysis" title="Glossary of functional analysis">glossary</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Spaces</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/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 distance</a></li> <li><a href="/wiki/Tomita%E2%80%93Takesaki_theory" title="Tomita–Takesaki theory">Tomita–Takesaki theory</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Functional_analysis" title="Category:Functional analysis">Category</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="Hilbert_spaces69" 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:Hilbert_space" title="Template:Hilbert space"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Hilbert_space" title="Template talk:Hilbert space"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Hilbert_space" title="Special:EditPage/Template:Hilbert space"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Hilbert_spaces69" style="font-size:114%;margin:0 4em"><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert 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/Hermitian_adjoint" title="Hermitian adjoint">Adjoint</a></li> <li><a class="mw-selflink selflink">Inner product</a> and <a href="/wiki/L-semi-inner_product" title="L-semi-inner product">L-semi-inner product</a></li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> and <a href="/wiki/Prehilbert_space" class="mw-redirect" title="Prehilbert space">Prehilbert space</a></li> <li><a href="/wiki/Orthogonal_complement" title="Orthogonal complement">Orthogonal complement</a></li> <li><a href="/wiki/Orthonormal_basis" title="Orthonormal basis">Orthonormal basis</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results</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/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/Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation</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"> <ul><li><a href="/wiki/Hilbert_projection_theorem" title="Hilbert projection theorem">Hilbert projection theorem</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> (<a href="/wiki/Parallelogram_law#The_parallelogram_law_in_inner_product_spaces" title="Parallelogram law">Parallelogram law</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-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Compact_operator_on_Hilbert_space" title="Compact operator on Hilbert space">Compact operator on Hilbert space</a></li> <li><a href="/wiki/Densely_defined_operator" title="Densely defined operator">Densely defined</a></li> <li><a href="/wiki/Sesquilinear_form#Hermitian_form" title="Sesquilinear form">Hermitian form</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/Self-adjoint_operator" title="Self-adjoint operator">Self-adjoint</a></li> <li><a href="/wiki/Sesquilinear_form" title="Sesquilinear form">Sesquilinear form</a></li> <li><a href="/wiki/Trace_class" title="Trace class">Trace class</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%">Examples</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/Distribution_(mathematics)" title="Distribution (mathematics)"><i>C</i><sup><i>n</i></sup>(<i>K</i>) with <i>K</i> compact & <i>n</i><∞</a></li> <li><a href="/wiki/Segal%E2%80%93Bargmann_space" title="Segal–Bargmann space">Segal–Bargmann <i>F</i></a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div 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Inner product space - Wikipedia