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id="toc-Ursescu_theorem-sublist" class="vector-toc-list"> <li id="toc-Statement" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Statement"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Statement</span> </div> </a> <ul id="toc-Statement-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Corollaries" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Corollaries"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Corollaries</span> </div> </a> <button aria-controls="toc-Corollaries-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 Corollaries subsection</span> </button> <ul id="toc-Corollaries-sublist" class="vector-toc-list"> <li id="toc-Closed_graph_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Closed_graph_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Closed graph theorem</span> </div> </a> <ul id="toc-Closed_graph_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uniform_boundedness_principle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uniform_boundedness_principle"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Uniform boundedness principle</span> </div> </a> <ul id="toc-Uniform_boundedness_principle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Open_mapping_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Open_mapping_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Open mapping theorem</span> </div> </a> <ul id="toc-Open_mapping_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Additional_corollaries" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Additional_corollaries"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Additional corollaries</span> </div> </a> <ul id="toc-Additional_corollaries-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Related_theorems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_theorems"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Related theorems</span> </div> </a> <button aria-controls="toc-Related_theorems-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 Related theorems subsection</span> </button> <ul id="toc-Related_theorems-sublist" class="vector-toc-list"> <li id="toc-Simons'_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simons'_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Simons' theorem</span> </div> </a> <ul id="toc-Simons'_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Robinson–Ursescu_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Robinson–Ursescu_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Robinson–Ursescu theorem</span> </div> </a> <ul id="toc-Robinson–Ursescu_theorem-sublist" class="vector-toc-list"> </ul> </li> </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">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" 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href="/wiki/Closed_graph_theorem_(functional_analysis)" title="Closed graph theorem (functional analysis)">closed graph theorem</a>, the <a href="/wiki/Open_mapping_theorem_(functional_analysis)" title="Open mapping theorem (functional analysis)">open mapping theorem</a>, and the <a href="/wiki/Uniform_boundedness_principle" title="Uniform boundedness principle">uniform boundedness principle</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Ursescu_theorem">Ursescu theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ursescu_theorem&action=edit&section=1" title="Edit section: Ursescu theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following notation and notions are used, 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 {\mathcal {R}}:X\rightrightarrows Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo>:</mo> <mi>X</mi> <mo stretchy="false">⇉</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}:X\rightrightarrows Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6a15a941b88ba5018a42f63a911b725b0a41893" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.275ex; height:2.343ex;" alt="{\displaystyle {\mathcal {R}}:X\rightrightarrows Y}" /></span> is a <a href="/wiki/Set-valued_function" title="Set-valued function">set-valued function</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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> is a non-empty subset of a <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector 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 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>: </p> <ul><li>the <a href="/wiki/Affine_span" class="mw-redirect" title="Affine span">affine span</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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> is 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 \operatorname {aff} S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>aff</mi> <mo>⁡</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {aff} S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e3802f165480db05aef385da6499e365e0b40c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.473ex; height:2.176ex;" alt="{\displaystyle \operatorname {aff} S}" /></span> and the <a href="/wiki/Linear_span" title="Linear span">linear span</a> is 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 \operatorname {span} S.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>span</mi> <mo>⁡</mo> <mi>S</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {span} S.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd86820720931a1d0e7ae8593d51224eff5eafd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.197ex; height:2.509ex;" alt="{\displaystyle \operatorname {span} S.}" /></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 S^{i}:=\operatorname {aint} _{X}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>:=</mo> <msub> <mi>aint</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{i}:=\operatorname {aint} _{X}S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b06d5060f1b61fd678bbd93f5848152df18418b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.592ex; height:3.009ex;" alt="{\displaystyle S^{i}:=\operatorname {aint} _{X}S}" /></span> denotes the <a href="/wiki/Algebraic_interior" title="Algebraic interior">algebraic interior</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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></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 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/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" /></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 {}^{i}S:=\operatorname {aint} _{\operatorname {aff} (S-S)}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mi>S</mi> <mo>:=</mo> <msub> <mi>aint</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>aff</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo>−</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}^{i}S:=\operatorname {aint} _{\operatorname {aff} (S-S)}S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79eb1fafd1262c700c5d869985cbc7a23deab3fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.676ex; height:3.509ex;" alt="{\displaystyle {}^{i}S:=\operatorname {aint} _{\operatorname {aff} (S-S)}S}" /></span> denotes the <a href="/wiki/Algebraic_interior#Relative_algebraic_interior" title="Algebraic interior">relative algebraic interior</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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> (i.e. the algebraic interior 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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></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 \operatorname {aff} (S-S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>aff</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo>−</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {aff} (S-S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e58cce369be56ef5bbda13f90a52f0732bbb1cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.234ex; height:2.843ex;" alt="{\displaystyle \operatorname {aff} (S-S)}" /></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 {}^{ib}S:={}^{i}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>b</mi> </mrow> </msup> <mi>S</mi> <mo>:=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}^{ib}S:={}^{i}S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f05f766adb0f8b90c682d2037f0966cd106d656" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.049ex; height:2.676ex;" alt="{\displaystyle {}^{ib}S:={}^{i}S}" /></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 \operatorname {span} \left(S-s_{0}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>span</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>S</mi> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {span} \left(S-s_{0}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53f0cd4135917e6835c7cc8ce27d8ed5ca931400" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.957ex; height:2.843ex;" alt="{\displaystyle \operatorname {span} \left(S-s_{0}\right)}" /></span> is <a href="/wiki/Barrelled_space" title="Barrelled space">barreled</a> for some/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 s_{0}\in S}"> <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> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{0}\in S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58a47244e0e0d4efe4a266ac1caeab46d72e931f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.485ex; height:2.509ex;" alt="{\displaystyle s_{0}\in S}" /></span> while <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}^{ib}S:=\varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>b</mi> </mrow> </msup> <mi>S</mi> <mo>:=</mo> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}^{ib}S:=\varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/117ecc0dc2f393570ff16562625aabec5e407789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.558ex; height:2.676ex;" alt="{\displaystyle {}^{ib}S:=\varnothing }" /></span> otherwise. <ul><li>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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> is convex then it can be shown that 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\in X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ebdb0a09f0721ccdd0b779e0a21caf386be82a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}" /></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 x\in {}^{ib}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>b</mi> </mrow> </msup> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in {}^{ib}S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6cae26133c030ab6ebec31e539ea46385755d45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.175ex; height:2.676ex;" alt="{\displaystyle x\in {}^{ib}S}" /></span> if and only if the cone generated 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 S-x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>−</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S-x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/121f9c68184b954bf6cbedb8e4babdf8f3dab9a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.669ex; height:2.343ex;" alt="{\displaystyle S-x}" /></span> is a barreled linear 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 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> or equivalently, 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 \cup _{n\in \mathbb {N} }n(S-x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <mi>n</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cup _{n\in \mathbb {N} }n(S-x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f7a3236fa212cea6521c302ca65df7ab7624d09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.925ex; height:2.843ex;" alt="{\displaystyle \cup _{n\in \mathbb {N} }n(S-x)}" /></span> is a barreled linear 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 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></li></ul></li> <li>The <b>domain 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 {\mathcal {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\displaystyle {\mathcal {R}}}" /></span></b> 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 \operatorname {Dom} {\mathcal {R}}:=\{x\in X:{\mathcal {R}}(x)\neq \varnothing \}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Dom</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mi class="MJX-variant">∅</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Dom} {\mathcal {R}}:=\{x\in X:{\mathcal {R}}(x)\neq \varnothing \}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/819c52b0aa988db4d69e9709db1d3d0af1c8e145" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.053ex; height:2.843ex;" alt="{\displaystyle \operatorname {Dom} {\mathcal {R}}:=\{x\in X:{\mathcal {R}}(x)\neq \varnothing \}.}" /></span></li> <li>The <b>image 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 {\mathcal {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\displaystyle {\mathcal {R}}}" /></span></b> 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 \operatorname {Im} {\mathcal {R}}:=\cup _{x\in X}{\mathcal {R}}(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Im</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo>:=</mo> <msub> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Im} {\mathcal {R}}:=\cup _{x\in X}{\mathcal {R}}(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abbce92222dc99cf9c8d364685151957a6818474" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.854ex; height:2.843ex;" alt="{\displaystyle \operatorname {Im} {\mathcal {R}}:=\cup _{x\in X}{\mathcal {R}}(x).}" /></span> For any 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 A\subseteq X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8a030c36801bc01950928058517e7a7da7eaa08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.468ex; height:2.509ex;" alt="{\displaystyle A\subseteq X,}" /></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 {\mathcal {R}}(A):=\cup _{x\in A}{\mathcal {R}}(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <msub> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}(A):=\cup _{x\in A}{\mathcal {R}}(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1561584d48fe547faa6c9ea070c74acfbb8d7915" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.077ex; height:2.843ex;" alt="{\displaystyle {\mathcal {R}}(A):=\cup _{x\in A}{\mathcal {R}}(x).}" /></span></li> <li>The <b>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 {\mathcal {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\displaystyle {\mathcal {R}}}" /></span></b> 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 \operatorname {gr} {\mathcal {R}}:=\{(x,y)\in X\times Y:y\in {\mathcal {R}}(x)\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>gr</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>X</mi> <mo>×</mo> <mi>Y</mi> <mo>:</mo> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {gr} {\mathcal {R}}:=\{(x,y)\in X\times Y:y\in {\mathcal {R}}(x)\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c267030116c33431d13f7034946b5768d2767362" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.955ex; height:2.843ex;" alt="{\displaystyle \operatorname {gr} {\mathcal {R}}:=\{(x,y)\in X\times Y:y\in {\mathcal {R}}(x)\}.}" /></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 {\mathcal {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\displaystyle {\mathcal {R}}}" /></span> is <b>closed</b> (respectively, <b>convex</b>) if 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 {\mathcal {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\displaystyle {\mathcal {R}}}" /></span> is closed (resp. convex) 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 X\times 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\times Y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41f39ef78be28dc2b9015ff7f82e9a1ef719a9f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.241ex; height:2.176ex;" alt="{\displaystyle X\times Y.}" /></span> <ul><li>Note 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 {\mathcal {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\displaystyle {\mathcal {R}}}" /></span> is convex 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_{0},x_{1}\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0},x_{1}\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72ca6672e54cf88f832cd0a605d27c4e55cb38e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.622ex; height:2.509ex;" alt="{\displaystyle x_{0},x_{1}\in X}" /></span> and 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 r\in [0,1],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\in [0,1],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89b836314fe509bfba3388b1d25eda2e1f9f85b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.189ex; height:2.843ex;" alt="{\displaystyle r\in [0,1],}" /></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 r{\mathcal {R}}\left(x_{0}\right)+(1-r){\mathcal {R}}\left(x_{1}\right)\subseteq {\mathcal {R}}\left(rx_{0}+(1-r)x_{1}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r{\mathcal {R}}\left(x_{0}\right)+(1-r){\mathcal {R}}\left(x_{1}\right)\subseteq {\mathcal {R}}\left(rx_{0}+(1-r)x_{1}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef5b087724afbc62d97cac086fbc15bb23fbdd95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.669ex; height:2.843ex;" alt="{\displaystyle r{\mathcal {R}}\left(x_{0}\right)+(1-r){\mathcal {R}}\left(x_{1}\right)\subseteq {\mathcal {R}}\left(rx_{0}+(1-r)x_{1}\right).}" /></span></li></ul></li> <li>The <b>inverse 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 {\mathcal {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\displaystyle {\mathcal {R}}}" /></span></b> is the set-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 {\mathcal {R}}^{-1}:Y\rightrightarrows X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>:</mo> <mi>Y</mi> <mo stretchy="false">⇉</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}^{-1}:Y\rightrightarrows X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d98695ee88b80e1f970972954768f4ca5e39065" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.608ex; height:2.843ex;" alt="{\displaystyle {\mathcal {R}}^{-1}:Y\rightrightarrows X}" /></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 {\mathcal {R}}^{-1}(y):=\{x\in X:y\in {\mathcal {R}}(x)\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}^{-1}(y):=\{x\in X:y\in {\mathcal {R}}(x)\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/381757f14672fa07242a0e96bcc3cf20246028d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.179ex; height:3.176ex;" alt="{\displaystyle {\mathcal {R}}^{-1}(y):=\{x\in X:y\in {\mathcal {R}}(x)\}.}" /></span> For any 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 B\subseteq Y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊆</mo> <mi>Y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\subseteq Y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a69adc349545e2a20f31eb474960db56ccd737b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.283ex; height:2.509ex;" alt="{\displaystyle B\subseteq Y,}" /></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 {\mathcal {R}}^{-1}(B):=\cup _{y\in B}{\mathcal {R}}^{-1}(y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <msub> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>∈</mo> <mi>B</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}^{-1}(B):=\cup _{y\in B}{\mathcal {R}}^{-1}(y).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbfae1aa0e0590057a4873f1022da9fd89db5a0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.48ex; height:3.343ex;" alt="{\displaystyle {\mathcal {R}}^{-1}(B):=\cup _{y\in B}{\mathcal {R}}^{-1}(y).}" /></span> <ul><li>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 f:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" /></span> is a function, then its inverse is the set-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^{-1}:Y\rightrightarrows X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>:</mo> <mi>Y</mi> <mo stretchy="false">⇉</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}:Y\rightrightarrows X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47ae2efcbd2db851229146b0afeb5faa2e5ed04f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.958ex; height:3.009ex;" alt="{\displaystyle f^{-1}:Y\rightrightarrows X}" /></span> obtained from canonically identifying <span class="mwe-math-element"><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 the set-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:X\rightrightarrows Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">⇉</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\rightrightarrows Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cde8858e2ba578a8976c1a40982a9dce64829e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\rightrightarrows Y}" /></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 x\mapsto \{f(x)\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto \{f(x)\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9036e9e1b33ec6c3d8645d7dfe7873905bbc9573" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.333ex; height:2.843ex;" alt="{\displaystyle x\mapsto \{f(x)\}.}" /></span></li></ul></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 \operatorname {int} _{T}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {int} _{T}S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47ffaae783f3d1cfcab5fd3d86cc75b82a286b9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.12ex; height:2.509ex;" alt="{\displaystyle \operatorname {int} _{T}S}" /></span> is the <a href="/wiki/Interior_(topology)" title="Interior (topology)">topological interior</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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> with respect 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 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/30ee86b741f542feb8f95f3c81fd53b043a25e26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.283ex; height:2.509ex;" alt="{\displaystyle T,}" /></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 S\subseteq T.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>T</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq T.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4366c3495b90f90f1635f89d7b0b01600c2e0da9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.881ex; height:2.343ex;" alt="{\displaystyle S\subseteq T.}" /></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 \operatorname {rint} S:=\operatorname {int} _{\operatorname {aff} S}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>rint</mi> <mo>⁡</mo> <mi>S</mi> <mo>:=</mo> <msub> <mi>int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>aff</mi> <mo>⁡</mo> <mi>S</mi> </mrow> </msub> <mo>⁡</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {rint} S:=\operatorname {int} _{\operatorname {aff} S}S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b5e64644d3d19affa2b6e333207d3b45376d99f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.626ex; height:2.509ex;" alt="{\displaystyle \operatorname {rint} S:=\operatorname {int} _{\operatorname {aff} S}S}" /></span> is the <a href="/wiki/Interior_(topology)" title="Interior (topology)">interior</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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> with respect 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 {aff} S.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>aff</mi> <mo>⁡</mo> <mi>S</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {aff} S.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a67c375ffac3f05130c94355898177a37e71f00e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.119ex; height:2.176ex;" alt="{\displaystyle \operatorname {aff} S.}" /></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Statement">Statement</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ursescu_theorem&action=edit&section=2" title="Edit section: Statement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1110004140">.mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa;overflow:hidden}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em}}</style><div class="math_theorem" style=""> <p><strong class="theorem-name">Theorem<sup id="cite_ref-FOOTNOTEZălinescu200223_1-0" class="reference"><a href="#cite_note-FOOTNOTEZălinescu200223-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></strong> <span class="theorem-note">(<b>Ursescu</b>)</span><span class="theoreme-tiret"> — </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 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> be a <a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">complete</a> <a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">semi-metrizable</a> <a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">locally convex</a> <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector space</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 {\mathcal {R}}:X\rightrightarrows Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo>:</mo> <mi>X</mi> <mo stretchy="false">⇉</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}:X\rightrightarrows Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6a15a941b88ba5018a42f63a911b725b0a41893" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.275ex; height:2.343ex;" alt="{\displaystyle {\mathcal {R}}:X\rightrightarrows Y}" /></span> be a <a href="/wiki/Closed_(mathematics)" class="mw-redirect" title="Closed (mathematics)">closed</a> <a href="/wiki/Convex_function" title="Convex function">convex</a> multifunction with non-empty domain. Assume 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 \operatorname {span} (\operatorname {Im} {\mathcal {R}}-y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>span</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Im</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {span} (\operatorname {Im} {\mathcal {R}}-y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e18b15a8c1a052894cebef4146bc63ee5dac207" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.602ex; height:2.843ex;" alt="{\displaystyle \operatorname {span} (\operatorname {Im} {\mathcal {R}}-y)}" /></span> is a <a href="/wiki/Barrelled_space" title="Barrelled space">barrelled space</a> for some/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 y\in \operatorname {Im} {\mathcal {R}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>Im</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in \operatorname {Im} {\mathcal {R}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfc5e91371b36e57cb3095bc87ff8ac25c9a96f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.776ex; height:2.509ex;" alt="{\displaystyle y\in \operatorname {Im} {\mathcal {R}}.}" /></span> Assume 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 y_{0}\in {}^{i}(\operatorname {Im} {\mathcal {R}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>Im</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{0}\in {}^{i}(\operatorname {Im} {\mathcal {R}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bee7bd89688d84e404590e10739b23c1ca7fc235" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.776ex; height:3.176ex;" alt="{\displaystyle y_{0}\in {}^{i}(\operatorname {Im} {\mathcal {R}})}" /></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 x_{0}\in {\mathcal {R}}^{-1}\left(y_{0}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}\in {\mathcal {R}}^{-1}\left(y_{0}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97fff88c344809041fc7e5fe6b58dc1e801d41ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.918ex; height:3.176ex;" alt="{\displaystyle x_{0}\in {\mathcal {R}}^{-1}\left(y_{0}\right)}" /></span> (so 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 y_{0}\in {\mathcal {R}}\left(x_{0}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{0}\in {\mathcal {R}}\left(x_{0}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8d613de162eabfe020f12cdd068febf4b56ad14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.585ex; height:2.843ex;" alt="{\displaystyle y_{0}\in {\mathcal {R}}\left(x_{0}\right)}" /></span>). Then for every neighborhood <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" /></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 x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </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/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" /></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 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/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" /></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 y_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d943dbbb0b56ca750c4d62c5b54b4ae29a773da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{0}}" /></span> belongs to the relative interior 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 {\mathcal {R}}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}(U)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f79bb3122659cb62c5156a11c9ae739e30edb97a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.563ex; height:2.843ex;" alt="{\displaystyle {\mathcal {R}}(U)}" /></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 \operatorname {aff} (\operatorname {Im} {\mathcal {R}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>aff</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Im</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {aff} (\operatorname {Im} {\mathcal {R}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fd23273489c08f457a93eb5161bb42d2d097478" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.529ex; height:2.843ex;" alt="{\displaystyle \operatorname {aff} (\operatorname {Im} {\mathcal {R}})}" /></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 y_{0}\in \operatorname {int} _{\operatorname {aff} (\operatorname {Im} {\mathcal {R}})}{\mathcal {R}}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <msub> <mi>int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>aff</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Im</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{0}\in \operatorname {int} _{\operatorname {aff} (\operatorname {Im} {\mathcal {R}})}{\mathcal {R}}(U)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a972dfdd14fbc6dedbf4848fef1523287dd22c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.911ex; height:3.176ex;" alt="{\displaystyle y_{0}\in \operatorname {int} _{\operatorname {aff} (\operatorname {Im} {\mathcal {R}})}{\mathcal {R}}(U)}" /></span>). In particular, 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 {}^{ib}(\operatorname {Im} {\mathcal {R}})\neq \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>b</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>Im</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>≠</mo> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})\neq \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf7fd2ebc86190d3c4cce6f1d3eda9cfbee629da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.354ex; height:3.176ex;" alt="{\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})\neq \varnothing }" /></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 {}^{ib}(\operatorname {Im} {\mathcal {R}})={}^{i}(\operatorname {Im} {\mathcal {R}})=\operatorname {rint} (\operatorname {Im} {\mathcal {R}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>b</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>Im</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>Im</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>rint</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Im</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})={}^{i}(\operatorname {Im} {\mathcal {R}})=\operatorname {rint} (\operatorname {Im} {\mathcal {R}}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e926526fee89b2a10fccc4e9c0271f1b699d1679" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.732ex; height:3.176ex;" alt="{\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})={}^{i}(\operatorname {Im} {\mathcal {R}})=\operatorname {rint} (\operatorname {Im} {\mathcal {R}}).}" /></span> </p> </div> <div class="mw-heading mw-heading2"><h2 id="Corollaries">Corollaries</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ursescu_theorem&action=edit&section=3" title="Edit section: Corollaries"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Closed_graph_theorem">Closed graph theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ursescu_theorem&action=edit&section=4" title="Edit section: Closed graph theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140" /><div class="math_theorem" style=""> <p><strong class="theorem-name"><a href="/wiki/Closed_graph_theorem" title="Closed graph theorem">Closed graph theorem</a></strong><span class="theoreme-tiret"> — </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 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> </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/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" /></span> be <a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet spaces</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 T:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T:X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d961a4704b054500d6ecff6b4caac6852982458c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.941ex; height:2.176ex;" alt="{\displaystyle T:X\to Y}" /></span> be a linear map. 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 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> is continuous if and only if 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> </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> is closed 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 X\times 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\times Y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41f39ef78be28dc2b9015ff7f82e9a1ef719a9f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.241ex; height:2.176ex;" alt="{\displaystyle X\times Y.}" /></span> </p> </div> <style data-mw-deduplicate="TemplateStyles:r1174254338">.mw-parser-output .math_proof{border:thin solid #aaa;margin:1em 2em;padding:0.5em 1em 0.4em}@media(max-width:500px){.mw-parser-output .math_proof{margin:1em 0;padding:0.5em 0.5em 0.4em}}</style><div class="math_proof" style=""><strong>Proof</strong> <p>For the non-trivial direction, assume that 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> </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> is closed 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 {\mathcal {R}}:=T^{-1}:Y\rightrightarrows X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo>:=</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>:</mo> <mi>Y</mi> <mo stretchy="false">⇉</mo> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}:=T^{-1}:Y\rightrightarrows X.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ad270c07faccf824122e18328c974652d9ec7b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.72ex; height:2.843ex;" alt="{\displaystyle {\mathcal {R}}:=T^{-1}:Y\rightrightarrows X.}" /></span> It is easy to see 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 \operatorname {gr} {\mathcal {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>gr</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {gr} {\mathcal {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e57c60a63ba36ebdcd0c5847dfbb7879ef07122" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.432ex; height:2.509ex;" alt="{\displaystyle \operatorname {gr} {\mathcal {R}}}" /></span> is closed and convex and that its image 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.}"> <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/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" /></span> Given <span class="mwe-math-element"><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 X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ebdb0a09f0721ccdd0b779e0a21caf386be82a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}" /></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 (Tx,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>T</mi> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Tx,x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9632cddd09275c1c8cf16bc9dc7608bee8f5e3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.139ex; height:2.843ex;" alt="{\displaystyle (Tx,x)}" /></span> 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 Y\times X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>×</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\times X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46759cca4940bbda4dc98d24fc0a5a8cd16d17f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.594ex; height:2.176ex;" alt="{\displaystyle Y\times X}" /></span> so that for every open neighborhood <span class="mwe-math-element"><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> 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 Tx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Tx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1898a2e49f896c6e60a294995b426cef8a13228" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.966ex; height:2.176ex;" alt="{\displaystyle Tx}" /></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 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> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}(V)=T^{-1}(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}(V)=T^{-1}(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf81ca8f99ad50ea20dea8e5c790e849032e9ad8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.315ex; height:3.176ex;" alt="{\displaystyle {\mathcal {R}}(V)=T^{-1}(V)}" /></span> is a neighborhood 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> 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 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/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" /></span> Thus <span class="mwe-math-element"><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> is continuous at <span class="mwe-math-element"><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> <a href="/wiki/Q.E.D." title="Q.E.D.">Q.E.D.</a> </p> </div> <div class="mw-heading mw-heading3"><h3 id="Uniform_boundedness_principle">Uniform boundedness principle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ursescu_theorem&action=edit&section=5" title="Edit section: Uniform boundedness principle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140" /><div class="math_theorem" style=""> <p><strong class="theorem-name"><a href="/wiki/Uniform_boundedness_principle" title="Uniform boundedness principle">Uniform boundedness principle</a></strong><span class="theoreme-tiret"> — </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 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> </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/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" /></span> be <a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet spaces</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 T:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T:X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d961a4704b054500d6ecff6b4caac6852982458c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.941ex; height:2.176ex;" alt="{\displaystyle T:X\to Y}" /></span> be a bijective linear map. 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 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> is continuous 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 T^{-1}:Y\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>:</mo> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{-1}:Y\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d6a8c7204ed8ee15e45c3aeba8c832985dd14ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.357ex; height:2.676ex;" alt="{\displaystyle T^{-1}:Y\to X}" /></span> is continuous. 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 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> is continuous 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 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> is an isomorphism of <a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet spaces</a>. </p> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338" /><div class="math_proof" style=""><strong>Proof</strong> <p>Apply the closed graph theorem 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 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> 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 T^{-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{-1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85ec6adde2d40e9a29246261dbb476b0e1095cb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.7ex; height:2.676ex;" alt="{\displaystyle T^{-1}.}" /></span> Q.E.D. </p> </div> <div class="mw-heading mw-heading3"><h3 id="Open_mapping_theorem">Open mapping theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ursescu_theorem&action=edit&section=6" title="Edit section: Open mapping theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140" /><div class="math_theorem" style=""> <p><strong class="theorem-name"><a href="/wiki/Open_mapping_theorem_(functional_analysis)" title="Open mapping theorem (functional analysis)">Open mapping theorem</a></strong><span class="theoreme-tiret"> — </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 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> </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/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" /></span> be <a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet spaces</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 T:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T:X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d961a4704b054500d6ecff6b4caac6852982458c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.941ex; height:2.176ex;" alt="{\displaystyle T:X\to Y}" /></span> be a continuous surjective linear map. Then T is an <a href="/wiki/Open_map" class="mw-redirect" title="Open map">open map</a>. </p> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338" /><div class="math_proof" style=""><strong>Proof</strong> <p>Clearly, <span class="mwe-math-element"><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> is a closed and convex relation whose image 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 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/0c668649af47a30006f93c9847d61fee8d9ffb61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}" /></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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" /></span> be a non-empty open subset 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/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" /></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 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> be 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 T(U),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(U),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90cb54a47a1459348a326b3a2958f2c2494728e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.875ex; height:2.843ex;" alt="{\displaystyle T(U),}" /></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 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 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" /></span> be 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 y=Tx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>T</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=Tx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b5e8122d563b930e71edd9c8cbef39b70f15473" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.867ex; height:2.509ex;" alt="{\displaystyle y=Tx.}" /></span> From the Ursescu theorem 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 T(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(U)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2fc49b0985e6766c1b4b70eb87b1e662888ac6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.228ex; height:2.843ex;" alt="{\displaystyle T(U)}" /></span> is a neighborhood 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> Q.E.D. </p> </div> <div class="mw-heading mw-heading3"><h3 id="Additional_corollaries">Additional corollaries</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ursescu_theorem&action=edit&section=7" title="Edit section: Additional corollaries"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following notation and notions are used for these corollaries, 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 {\mathcal {R}}:X\rightrightarrows Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo>:</mo> <mi>X</mi> <mo stretchy="false">⇉</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}:X\rightrightarrows Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6a15a941b88ba5018a42f63a911b725b0a41893" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.275ex; height:2.343ex;" alt="{\displaystyle {\mathcal {R}}:X\rightrightarrows Y}" /></span> is a set-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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> is a non-empty subset of a <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector 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 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>: </p> <ul><li>a <b><a href="/wiki/Convex_series" title="Convex series">convex series</a> with 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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span></b> is a <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a> of the 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="{\textstyle \sum _{i=1}^{\infty }r_{i}s_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" 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 mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i=1}^{\infty }r_{i}s_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c55558981df75be1140b9c7c7c912baaa7be6ac2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.48ex; height:3.176ex;" alt="{\textstyle \sum _{i=1}^{\infty }r_{i}s_{i}}" /></span> where 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 s_{i}\in S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{i}\in S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87a8178559b9f271a78c8287257e81fd924faef5" 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 s_{i}\in S}" /></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="{\textstyle \sum _{i=1}^{\infty }r_{i}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" 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 mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i=1}^{\infty }r_{i}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2addef3298ebd737855ba896d345d54007e248d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.85ex; height:3.176ex;" alt="{\textstyle \sum _{i=1}^{\infty }r_{i}=1}" /></span> is a series of non-negative numbers. 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="{\textstyle \sum _{i=1}^{\infty }r_{i}s_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" 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 mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i=1}^{\infty }r_{i}s_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c55558981df75be1140b9c7c7c912baaa7be6ac2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.48ex; height:3.176ex;" alt="{\textstyle \sum _{i=1}^{\infty }r_{i}s_{i}}" /></span> converges then the series is called <b>convergent</b> while 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 \left(s_{i}\right)_{i=1}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow> <mo>(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(s_{i}\right)_{i=1}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6412b66e4aea2bc3224c4da51d77fc2f8994a947" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.6ex; height:3.176ex;" alt="{\displaystyle \left(s_{i}\right)_{i=1}^{\infty }}" /></span> is bounded then the series is called <b>bounded</b> and <b>b-convex</b>.</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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> is <b>ideally convex</b> if any convergent b-convex series 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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> has its sum 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 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/23bbb1f0f6ebdfa78b4fed06049640f7386bb44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.146ex; height:2.176ex;" alt="{\displaystyle S.}" /></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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> is <b>lower ideally convex</b> if there exists a <a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet 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 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/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; 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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /></span> is equal to the projection onto <span class="mwe-math-element"><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> of some ideally convex subset <i>B</i> 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\times 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\times Y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41f39ef78be28dc2b9015ff7f82e9a1ef719a9f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.241ex; height:2.176ex;" alt="{\displaystyle X\times Y.}" /></span> Every ideally convex set is lower ideally convex.</li></ul> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140" /><div class="math_theorem" style=""> <p><strong class="theorem-name">Corollary</strong><span class="theoreme-tiret"> — </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 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> be a barreled <a href="/wiki/First_countable_space" class="mw-redirect" title="First countable space">first countable space</a> 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 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> be a subset 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/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" /></span> Then: </p> <ol><li>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 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> is lower ideally convex 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 C^{i}=\operatorname {int} C.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>=</mo> <mi>int</mi> <mo>⁡</mo> <mi>C</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{i}=\operatorname {int} C.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1a8ece6aa4805f2b0b671bcbdaacbb8af05f17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.34ex; height:2.676ex;" alt="{\displaystyle C^{i}=\operatorname {int} C.}" /></span></li> <li>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 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> is ideally convex 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 C^{i}=\operatorname {int} C=\operatorname {int} \left(\operatorname {cl} C\right)=\left(\operatorname {cl} C\right)^{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>=</mo> <mi>int</mi> <mo>⁡</mo> <mi>C</mi> <mo>=</mo> <mi>int</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>cl</mi> <mo>⁡</mo> <mi>C</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>cl</mi> <mo>⁡</mo> <mi>C</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{i}=\operatorname {int} C=\operatorname {int} \left(\operatorname {cl} C\right)=\left(\operatorname {cl} C\right)^{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81ed9036d21ea6a023ef41372377a48dbafe43d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.465ex; height:3.343ex;" alt="{\displaystyle C^{i}=\operatorname {int} C=\operatorname {int} \left(\operatorname {cl} C\right)=\left(\operatorname {cl} C\right)^{i}.}" /></span></li></ol> </div> <div class="mw-heading mw-heading2"><h2 id="Related_theorems">Related theorems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ursescu_theorem&action=edit&section=8" title="Edit section: Related theorems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Simons'_theorem"><span id="Simons.27_theorem"></span>Simons' theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ursescu_theorem&action=edit&section=9" title="Edit section: Simons' theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140" /><div class="math_theorem" style=""> <p><strong class="theorem-name"><a href="/wiki/Jim_Simons" title="Jim Simons">Simons'</a> theorem<sup id="cite_ref-FOOTNOTEZălinescu200222-23_2-0" class="reference"><a href="#cite_note-FOOTNOTEZălinescu200222-23-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></strong><span class="theoreme-tiret"> — </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 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> </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/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" /></span> be <a href="/wiki/First_countable" class="mw-redirect" title="First countable">first countable</a> 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 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> locally convex. 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 {\mathcal {R}}:X\rightrightarrows Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo>:</mo> <mi>X</mi> <mo stretchy="false">⇉</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}:X\rightrightarrows Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6a15a941b88ba5018a42f63a911b725b0a41893" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.275ex; height:2.343ex;" alt="{\displaystyle {\mathcal {R}}:X\rightrightarrows Y}" /></span> is a multimap with non-empty domain that satisfies <a href="/wiki/Convex_series#Conditions_(Hx)_and_(Hwx)" title="Convex series">condition (Hw<i>x</i>)</a> or else assume 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}"> <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> is a <a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet space</a> and 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 {\mathcal {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\displaystyle {\mathcal {R}}}" /></span> is <a href="/wiki/Convex_series#Types_of_subsets" title="Convex series">lower ideally convex</a>. Assume 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 \operatorname {span} (\operatorname {Im} {\mathcal {R}}-y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>span</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Im</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {span} (\operatorname {Im} {\mathcal {R}}-y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e18b15a8c1a052894cebef4146bc63ee5dac207" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.602ex; height:2.843ex;" alt="{\displaystyle \operatorname {span} (\operatorname {Im} {\mathcal {R}}-y)}" /></span> is <a href="/wiki/Barrelled_space" title="Barrelled space">barreled</a> for some/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 y\in \operatorname {Im} {\mathcal {R}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>Im</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in \operatorname {Im} {\mathcal {R}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfc5e91371b36e57cb3095bc87ff8ac25c9a96f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.776ex; height:2.509ex;" alt="{\displaystyle y\in \operatorname {Im} {\mathcal {R}}.}" /></span> Assume 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 y_{0}\in {}^{i}(\operatorname {Im} {\mathcal {R}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>Im</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{0}\in {}^{i}(\operatorname {Im} {\mathcal {R}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bee7bd89688d84e404590e10739b23c1ca7fc235" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.776ex; height:3.176ex;" alt="{\displaystyle y_{0}\in {}^{i}(\operatorname {Im} {\mathcal {R}})}" /></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 x_{0}\in {\mathcal {R}}^{-1}\left(y_{0}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}\in {\mathcal {R}}^{-1}\left(y_{0}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b609f529727e73509aa8c91a51981f7f5beb2300" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.952ex; height:3.176ex;" alt="{\displaystyle x_{0}\in {\mathcal {R}}^{-1}\left(y_{0}\right).}" /></span> Then for every neighborhood <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" /></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 x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </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/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" /></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 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/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" /></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 y_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d943dbbb0b56ca750c4d62c5b54b4ae29a773da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{0}}" /></span> belongs to the relative interior 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 {\mathcal {R}}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}(U)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f79bb3122659cb62c5156a11c9ae739e30edb97a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.563ex; height:2.843ex;" alt="{\displaystyle {\mathcal {R}}(U)}" /></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 \operatorname {aff} (\operatorname {Im} {\mathcal {R}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>aff</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Im</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {aff} (\operatorname {Im} {\mathcal {R}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fd23273489c08f457a93eb5161bb42d2d097478" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.529ex; height:2.843ex;" alt="{\displaystyle \operatorname {aff} (\operatorname {Im} {\mathcal {R}})}" /></span> (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 y_{0}\in \operatorname {int} _{\operatorname {aff} (\operatorname {Im} {\mathcal {R}})}{\mathcal {R}}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <msub> <mi>int</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>aff</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Im</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{0}\in \operatorname {int} _{\operatorname {aff} (\operatorname {Im} {\mathcal {R}})}{\mathcal {R}}(U)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a972dfdd14fbc6dedbf4848fef1523287dd22c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.911ex; height:3.176ex;" alt="{\displaystyle y_{0}\in \operatorname {int} _{\operatorname {aff} (\operatorname {Im} {\mathcal {R}})}{\mathcal {R}}(U)}" /></span>). In particular, 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 {}^{ib}(\operatorname {Im} {\mathcal {R}})\neq \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>b</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>Im</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>≠</mo> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})\neq \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf7fd2ebc86190d3c4cce6f1d3eda9cfbee629da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.354ex; height:3.176ex;" alt="{\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})\neq \varnothing }" /></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 {}^{ib}(\operatorname {Im} {\mathcal {R}})={}^{i}(\operatorname {Im} {\mathcal {R}})=\operatorname {rint} (\operatorname {Im} {\mathcal {R}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>b</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>Im</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>Im</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>rint</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Im</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})={}^{i}(\operatorname {Im} {\mathcal {R}})=\operatorname {rint} (\operatorname {Im} {\mathcal {R}}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e926526fee89b2a10fccc4e9c0271f1b699d1679" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.732ex; height:3.176ex;" alt="{\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})={}^{i}(\operatorname {Im} {\mathcal {R}})=\operatorname {rint} (\operatorname {Im} {\mathcal {R}}).}" /></span> </p> </div> <div class="mw-heading mw-heading3"><h3 id="Robinson–Ursescu_theorem"><span id="Robinson.E2.80.93Ursescu_theorem"></span>Robinson–Ursescu theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ursescu_theorem&action=edit&section=10" title="Edit section: Robinson–Ursescu theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The implication (1) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \implies }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace"></mspace> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \implies }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/913c2e89ea94dfa446f69b056d4bf505e01fcc5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.096ex; height:1.843ex;" alt="{\displaystyle \implies }" /></span> (2) in the following theorem is known as the Robinson–Ursescu theorem.<sup id="cite_ref-FOOTNOTEZălinescu200224_3-0" class="reference"><a href="#cite_note-FOOTNOTEZălinescu200224-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140" /><div class="math_theorem" style=""> <p><strong class="theorem-name">Robinson–Ursescu theorem<sup id="cite_ref-FOOTNOTEZălinescu200224_3-1" class="reference"><a href="#cite_note-FOOTNOTEZălinescu200224-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></strong><span class="theoreme-tiret"> — </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 (X,\|\,\cdot \,\|)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mspace width="thinmathspace"></mspace> <mo>⋅</mo> <mspace width="thinmathspace"></mspace> <mo fence="false" stretchy="false">‖</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\|\,\cdot \,\|)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15d5aff0f4f2ef4433fbb34b70e6cc92187db708" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.601ex; height:2.843ex;" alt="{\displaystyle (X,\|\,\cdot \,\|)}" /></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,\|\,\cdot \,\|)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mspace width="thinmathspace"></mspace> <mo>⋅</mo> <mspace width="thinmathspace"></mspace> <mo fence="false" stretchy="false">‖</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Y,\|\,\cdot \,\|)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04b6720f287300c0ad2b7682e8f07323d744a960" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.395ex; height:2.843ex;" alt="{\displaystyle (Y,\|\,\cdot \,\|)}" /></span> be <a href="/wiki/Normed_space" class="mw-redirect" title="Normed space">normed spaces</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 {\mathcal {R}}:X\rightrightarrows Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo>:</mo> <mi>X</mi> <mo stretchy="false">⇉</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}:X\rightrightarrows Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6a15a941b88ba5018a42f63a911b725b0a41893" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.275ex; height:2.343ex;" alt="{\displaystyle {\mathcal {R}}:X\rightrightarrows Y}" /></span> be a multimap with non-empty domain. 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 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/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" /></span> is a <a href="/wiki/Barreled_space" class="mw-redirect" title="Barreled space">barreled space</a>, 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 {\mathcal {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\displaystyle {\mathcal {R}}}" /></span> verifies condition <a href="/wiki/Convex_series#Conditions_(Hx)_and_(Hwx)" title="Convex series">condition (Hw<i>x</i>)</a>, and 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},y_{0})\in \operatorname {gr} {\mathcal {R}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>gr</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{0},y_{0})\in \operatorname {gr} {\mathcal {R}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aa90624e2f0b1bf41f70d39b67ee9158c90e0fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.34ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0})\in \operatorname {gr} {\mathcal {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 C_{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7243577bf16ea88abb546ddb7b150b477ba3cfd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.294ex; height:2.509ex;" alt="{\displaystyle C_{X}}" /></span> (resp. <span class="mwe-math-element"><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_{Y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{Y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07e82ffcc1e44e78efdb97205c5f1d4129f58c7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.148ex; height:2.509ex;" alt="{\displaystyle C_{Y}}" /></span>) denote the closed unit ball 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 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> (resp. <span class="mwe-math-element"><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/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" /></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 C_{X}=\{x\in X:\|x\|\leq 1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{X}=\{x\in X:\|x\|\leq 1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/616cd6f9052b0e5fa770e3afeb67be446d6305fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.721ex; height:2.843ex;" alt="{\displaystyle C_{X}=\{x\in X:\|x\|\leq 1\}}" /></span>). Then the following are equivalent: </p> <ol><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 y_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d943dbbb0b56ca750c4d62c5b54b4ae29a773da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{0}}" /></span> belongs to the <a href="/wiki/Algebraic_interior" title="Algebraic interior">algebraic interior</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 \operatorname {Im} {\mathcal {R}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Im</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Im} {\mathcal {R}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c454c5310c3c45482da885af68f1991794da3235" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.78ex; height:2.176ex;" alt="{\displaystyle \operatorname {Im} {\mathcal {R}}.}" /></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 y_{0}\in \operatorname {int} {\mathcal {R}}\left(x_{0}+C_{X}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mi>int</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{0}\in \operatorname {int} {\mathcal {R}}\left(x_{0}+C_{X}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb51da843e92e87af439aee8715e4dac06ce5eb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.985ex; height:2.843ex;" alt="{\displaystyle y_{0}\in \operatorname {int} {\mathcal {R}}\left(x_{0}+C_{X}\right).}" /></span></li> <li>There exists <span class="mwe-math-element"><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>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b567393bbf6974b07d583790802107e2d46963dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.025ex; height:2.176ex;" alt="{\displaystyle B>0}" /></span> such that 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 0\leq r\leq 1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>r</mi> <mo>≤</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq r\leq 1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0497a34e61bae5262baaaf2bab6eeb040a68bb40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.217ex; height:2.509ex;" alt="{\displaystyle 0\leq r\leq 1,}" /></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 y_{0}+BrC_{Y}\subseteq {\mathcal {R}}\left(x_{0}+rC_{X}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>B</mi> <mi>r</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>r</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{0}+BrC_{Y}\subseteq {\mathcal {R}}\left(x_{0}+rC_{X}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a6e8eb5dedab4a852cf520f0b0fbbcc35375299" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.861ex; height:2.843ex;" alt="{\displaystyle y_{0}+BrC_{Y}\subseteq {\mathcal {R}}\left(x_{0}+rC_{X}\right).}" /></span></li> <li>There exist <span class="mwe-math-element"><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/232a94dbd97f2ce670d4987c5ad8ad82b072861b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.004ex; height:2.176ex;" alt="{\displaystyle 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 B>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b567393bbf6974b07d583790802107e2d46963dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.025ex; height:2.176ex;" alt="{\displaystyle B>0}" /></span> such that 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 x_{0}+AC_{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>A</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in x_{0}+AC_{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d992b402bf01a7eae473915cac2a739df7278cc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.432ex; height:2.509ex;" alt="{\displaystyle x\in x_{0}+AC_{X}}" /></span> and 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 y\in y_{0}+AC_{Y},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>A</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in y_{0}+AC_{Y},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7012bf640383730a2f1fb68c6d73b11104ced496" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.568ex; height:2.509ex;" alt="{\displaystyle y\in y_{0}+AC_{Y},}" /></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 d\left(x,{\mathcal {R}}^{-1}(y)\right)\leq B\cdot d(y,{\mathcal {R}}(x)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>≤</mo> <mi>B</mi> <mo>⋅</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\left(x,{\mathcal {R}}^{-1}(y)\right)\leq B\cdot d(y,{\mathcal {R}}(x)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693c3fdd0a83f88b76e15315a5203556187d89a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.877ex; height:3.343ex;" alt="{\displaystyle d\left(x,{\mathcal {R}}^{-1}(y)\right)\leq B\cdot d(y,{\mathcal {R}}(x)).}" /></span></li> <li>There exists <span class="mwe-math-element"><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>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b567393bbf6974b07d583790802107e2d46963dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.025ex; height:2.176ex;" alt="{\displaystyle B>0}" /></span> such that 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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" /></span> and 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 y\in y_{0}+BC_{Y},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>B</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in y_{0}+BC_{Y},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e2a946fdb0740e458aaba6143cc565a727c5bb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.589ex; height:2.509ex;" alt="{\displaystyle y\in y_{0}+BC_{Y},}" /></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 d\left(x,{\mathcal {R}}^{-1}(y)\right)\leq {\frac {1+\left\|x-x_{0}\right\|}{B-\left\|y-y_{0}\right\|}}\cdot d(y,{\mathcal {R}}(x)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mrow> <mo symmetric="true">‖</mo> <mrow> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo symmetric="true">‖</mo> </mrow> </mrow> <mrow> <mi>B</mi> <mo>−</mo> <mrow> <mo symmetric="true">‖</mo> <mrow> <mi>y</mi> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo symmetric="true">‖</mo> </mrow> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\left(x,{\mathcal {R}}^{-1}(y)\right)\leq {\frac {1+\left\|x-x_{0}\right\|}{B-\left\|y-y_{0}\right\|}}\cdot d(y,{\mathcal {R}}(x)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b1b18427392da88d93dc7d37f963e49a403999a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43.068ex; height:6.509ex;" alt="{\displaystyle d\left(x,{\mathcal {R}}^{-1}(y)\right)\leq {\frac {1+\left\|x-x_{0}\right\|}{B-\left\|y-y_{0}\right\|}}\cdot d(y,{\mathcal {R}}(x)).}" /></span></li></ol> </div> <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=Ursescu_theorem&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Closed_graph_theorem" title="Closed graph theorem">Closed graph theorem</a> – Theorem relating continuity to graphs</li> <li><a href="/wiki/Closed_graph_theorem_(functional_analysis)" title="Closed graph theorem (functional analysis)">Closed graph theorem (functional analysis)</a> – Theorems connecting continuity to closure of graphs</li> <li><a href="/wiki/Open_mapping_theorem_(functional_analysis)" title="Open mapping theorem (functional analysis)">Open mapping theorem (functional analysis)</a> – Condition for a linear operator to be open</li> <li><a href="/wiki/Surjection_of_Fr%C3%A9chet_spaces" title="Surjection of Fréchet spaces">Surjection of Fréchet spaces</a> – Characterization of surjectivity</li> <li><a href="/wiki/Uniform_boundedness_principle" title="Uniform boundedness principle">Uniform boundedness principle</a> – Theorem stating that pointwise boundedness implies uniform boundedness</li> <li><a href="/wiki/Webbed_space" title="Webbed space">Webbed space</a> – Space where open mapping and closed graph theorems hold</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=Ursescu_theorem&action=edit&section=12" 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> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-FOOTNOTEZălinescu200223-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEZălinescu200223_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFZălinescu2002">Zălinescu 2002</a>, p. 23.</span> </li> <li id="cite_note-FOOTNOTEZălinescu200222-23-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEZălinescu200222-23_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFZălinescu2002">Zălinescu 2002</a>, p. 22-23.</span> </li> <li id="cite_note-FOOTNOTEZălinescu200224-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEZălinescu200224_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEZălinescu200224_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFZălinescu2002">Zălinescu 2002</a>, p. 24.</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=Ursescu_theorem&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><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="CITEREFZălinescu2002" class="citation book cs1"><a href="/w/index.php?title=Constantin_Z%C4%83linescu&action=edit&redlink=1" class="new" title="Constantin Zălinescu (page does not exist)">Zălinescu, Constantin</a> (30 July 2002). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/convexanalysisge00zali_934"><i>Convex Analysis in General Vector Spaces</i></a></span>. River Edge, N.J. London: <a href="/wiki/World_Scientific_Publishing" class="mw-redirect" title="World Scientific Publishing">World Scientific Publishing</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4488-15-0" title="Special:BookSources/978-981-4488-15-0"><bdi>978-981-4488-15-0</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1921556">1921556</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/285163112">285163112</a> – via <a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Convex+Analysis+in+General+Vector+Spaces&rft.place=River+Edge%2C+N.J.+London&rft.pub=World+Scientific+Publishing&rft.date=2002-07-30&rft_id=info%3Aoclcnum%2F285163112&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1921556%23id-name%3DMR&rft.isbn=978-981-4488-15-0&rft.aulast=Z%C4%83linescu&rft.aufirst=Constantin&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fconvexanalysisge00zali_934&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUrsescu+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBaggs1974" class="citation journal cs1">Baggs, Ivan (1974). <a rel="nofollow" class="external text" href="https://www.ams.org/">"Functions with a closed graph"</a>. <i>Proceedings of the American Mathematical Society</i>. <b>43</b> (2): <span class="nowrap">439–</span>442. <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/10.1090%2FS0002-9939-1974-0334132-8">10.1090/S0002-9939-1974-0334132-8</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9939">0002-9939</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+American+Mathematical+Society&rft.atitle=Functions+with+a+closed+graph&rft.volume=43&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E439-%3C%2Fspan%3E442&rft.date=1974&rft_id=info%3Adoi%2F10.1090%2FS0002-9939-1974-0334132-8&rft.issn=0002-9939&rft.aulast=Baggs&rft.aufirst=Ivan&rft_id=https%3A%2F%2Fwww.ams.org%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUrsescu+theorem" class="Z3988"></span></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 ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist 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id="Convex_analysis_and_variational_analysis174" style="font-size:114%;margin:0 4em"><a href="/wiki/Convex_analysis" title="Convex analysis">Convex analysis</a> and <a href="/wiki/Variational_analysis" title="Variational analysis">variational analysis</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/Convex_combination" title="Convex combination">Convex combination</a></li> <li><a href="/wiki/Convex_function" title="Convex function">Convex function</a></li> <li><a href="/wiki/Convex_set" title="Convex set">Convex set</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_convexity_topics" title="List of convexity topics">Topics (list)</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/Choquet_theory" title="Choquet theory">Choquet theory</a></li> <li><a href="/wiki/Convex_geometry" title="Convex geometry">Convex geometry</a></li> <li><a href="/wiki/Convex_metric_space" title="Convex metric space">Convex metric space</a></li> <li><a href="/wiki/Convex_optimization" title="Convex optimization">Convex optimization</a></li> <li><a href="/wiki/Duality_(optimization)" title="Duality (optimization)">Duality</a></li> <li><a href="/wiki/Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a></li> <li><a href="/wiki/Legendre_transformation" title="Legendre transformation">Legendre transformation</a></li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex topological vector space</a></li> <li><a href="/wiki/Simplex" title="Simplex">Simplex</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Convex_conjugate" title="Convex conjugate">Convex conjugate</a></li> <li><a href="/wiki/Concave_function" title="Concave function">Concave</a></li> <li>(<a href="/wiki/Closed_convex_function" title="Closed convex function">Closed</a></li> <li><a href="/wiki/K-convex_function" title="K-convex function">K-</a></li> <li><a href="/wiki/Logarithmically_convex_function" title="Logarithmically convex function">Logarithmically</a></li> <li><a href="/wiki/Proper_convex_function" title="Proper convex function">Proper</a></li> <li><a href="/wiki/Pseudoconvex_function" title="Pseudoconvex function">Pseudo-</a></li> <li><a href="/wiki/Quasiconvex_function" title="Quasiconvex function">Quasi-</a>) <a href="/wiki/Convex_function" title="Convex function">Convex function</a></li> <li><a href="/wiki/Invex_function" title="Invex function">Invex function</a></li> <li><a href="/wiki/Legendre_transformation" title="Legendre transformation">Legendre transformation</a></li> <li><a href="/wiki/Semi-continuity" title="Semi-continuity">Semi-continuity</a></li> <li><a href="/wiki/Subderivative" title="Subderivative">Subderivative</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main <a href="/wiki/Category:Theorems_involving_convexity" title="Category:Theorems involving convexity">results (list)</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Carath%C3%A9odory%27s_theorem_(convex_hull)" title="Carathéodory's theorem (convex hull)">Carathéodory's theorem</a></li> <li><a href="/wiki/Ekeland%27s_variational_principle" title="Ekeland's variational principle">Ekeland's variational principle</a></li> <li><a href="/wiki/Fenchel%E2%80%93Moreau_theorem" title="Fenchel–Moreau theorem">Fenchel–Moreau theorem</a></li> <li><a href="/wiki/Fenchel-Young_inequality" class="mw-redirect" title="Fenchel-Young inequality">Fenchel-Young inequality</a></li> <li><a href="/wiki/Jensen%27s_inequality" title="Jensen's inequality">Jensen's inequality</a></li> <li><a href="/wiki/Hermite%E2%80%93Hadamard_inequality" title="Hermite–Hadamard inequality">Hermite–Hadamard inequality</a></li> <li><a href="/wiki/Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman theorem</a></li> <li><a href="/wiki/Mazur%27s_lemma" title="Mazur's lemma">Mazur's lemma</a></li> <li><a href="/wiki/Shapley%E2%80%93Folkman_lemma" title="Shapley–Folkman lemma">Shapley–Folkman lemma</a></li> <li><a class="mw-selflink-fragment" href="#Robinson–Ursescu_theorem">Robinson–Ursescu</a></li> <li><a class="mw-selflink-fragment" href="#Simons'_theorem">Simons</a></li> <li><a class="mw-selflink selflink">Ursescu</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sets</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Convex_hull" title="Convex hull">Convex hull</a></li> <li>(<a href="/wiki/Orthogonally_convex_set" class="mw-redirect" title="Orthogonally convex set">Orthogonally</a>, <a href="/wiki/Pseudoconvexity" title="Pseudoconvexity">Pseudo-</a>) <a href="/wiki/Convex_set" title="Convex set">Convex set</a></li> <li><a href="/wiki/Effective_domain" title="Effective domain">Effective domain</a></li> <li><a href="/wiki/Epigraph_(mathematics)" title="Epigraph (mathematics)">Epigraph</a></li> <li><a href="/wiki/Hypograph_(mathematics)" title="Hypograph (mathematics)">Hypograph</a></li> <li><a href="/wiki/John_ellipsoid" title="John ellipsoid">John ellipsoid</a></li> <li><a href="/wiki/Lens_(geometry)" title="Lens (geometry)">Lens</a></li> <li><a href="/wiki/Radial_set" title="Radial set">Radial set</a>/<a href="/wiki/Algebraic_interior" title="Algebraic interior">Algebraic interior</a></li> <li><a href="/wiki/Zonotope" class="mw-redirect" title="Zonotope">Zonotope</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Series</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/Convex_series#Types_of_subsets" title="Convex series">Convex series related</a> (<a href="/wiki/Convex_series#Types_of_subsets" title="Convex series">(cs, lcs)-closed</a>, <a href="/wiki/Convex_series#Types_of_subsets" title="Convex series">(cs, bcs)-complete</a>, <a href="/wiki/Convex_series#Types_of_subsets" title="Convex series">(lower) ideally convex</a>, <a href="/wiki/Convex_series#Types_of_subsets" title="Convex series">(H<i>x</i>)</a>, and <a href="/wiki/Convex_series#Types_of_subsets" title="Convex series">(Hw<i>x</i>)</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Duality_(optimization)" title="Duality (optimization)">Duality</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/Dual_system" title="Dual system">Dual system</a></li> <li><a href="/wiki/Duality_gap" title="Duality gap">Duality gap</a></li> <li><a href="/wiki/Strong_duality" title="Strong duality">Strong duality</a></li> <li><a href="/wiki/Weak_duality" title="Weak duality">Weak duality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications and related</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/Convexity_in_economics" title="Convexity in economics">Convexity in economics</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="" 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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:Topological_vector_spaces" title="Template:Topological vector spaces"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Topological_vector_spaces" title="Template talk:Topological vector spaces"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Topological_vector_spaces" title="Special:EditPage/Template:Topological vector spaces"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Topological_vector_spaces_(TVSs)267" style="font-size:114%;margin:0 4em"><a href="/wiki/Topological_vector_space" title="Topological vector space">Topological vector spaces</a> (TVSs)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_space" title="Banach space">Banach space</a></li> <li><a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">Completeness</a></li> <li><a href="/wiki/Continuous_linear_operator" title="Continuous linear operator">Continuous linear operator</a></li> <li><a href="/wiki/Linear_form" title="Linear form">Linear functional</a></li> <li><a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet space</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a></li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex space</a></li> <li><a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">Metrizability</a></li> <li><a href="/wiki/Operator_topologies" title="Operator topologies">Operator topologies</a></li> <li><a href="/wiki/Topological_vector_space" title="Topological vector space">Topological vector space</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Theorems_in_functional_analysis" title="Category:Theorems in functional analysis">Main results</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Anderson%E2%80%93Kadec_theorem" title="Anderson–Kadec theorem">Anderson–Kadec</a></li> <li><a href="/wiki/Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu</a></li> <li><a href="/wiki/Closed_graph_theorem_(functional_analysis)" title="Closed graph theorem (functional analysis)">Closed graph theorem</a></li> <li><a href="/wiki/F._Riesz%27s_theorem" title="F. Riesz's theorem">F. Riesz's</a></li> <li><a href="/wiki/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a> (<a href="/wiki/Hyperplane_separation_theorem" title="Hyperplane separation theorem">hyperplane separation</a></li> <li><a href="/wiki/Vector-valued_Hahn%E2%80%93Banach_theorems" title="Vector-valued Hahn–Banach theorems">Vector-valued Hahn–Banach</a>)</li> <li><a href="/wiki/Open_mapping_theorem_(functional_analysis)" title="Open mapping theorem (functional analysis)">Open mapping (Banach–Schauder)</a> <ul><li><a href="/wiki/Bounded_inverse_theorem" class="mw-redirect" title="Bounded inverse theorem">Bounded inverse</a></li></ul></li> <li><a href="/wiki/Uniform_boundedness_principle" title="Uniform boundedness principle">Uniform boundedness (Banach–Steinhaus)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bilinear_operator" class="mw-redirect" title="Bilinear operator">Bilinear operator</a> <ul><li><a href="/wiki/Bilinear_form" title="Bilinear form">form</a></li></ul></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a> <ul><li><a href="/wiki/Almost_open_linear_map" class="mw-redirect" title="Almost open linear map">Almost open</a></li> <li><a href="/wiki/Bounded_operator" title="Bounded operator">Bounded</a></li> <li><a href="/wiki/Continuous_linear_operator" title="Continuous linear operator">Continuous</a></li> <li><a href="/wiki/Closed_linear_operator" title="Closed linear operator">Closed</a></li> <li><a href="/wiki/Compact_operator" title="Compact operator">Compact</a></li> <li><a href="/wiki/Densely_defined_operator" title="Densely defined operator">Densely defined</a></li> <li><a href="/wiki/Discontinuous_linear_map" title="Discontinuous linear map">Discontinuous</a></li></ul></li> <li><a href="/wiki/Topological_homomorphism" title="Topological homomorphism">Topological homomorphism</a></li> <li><a href="/wiki/Functional_(mathematics)" title="Functional (mathematics)">Functional</a> <ul><li><a href="/wiki/Linear_form" title="Linear form">Linear</a></li> <li><a href="/wiki/Bilinear_form" title="Bilinear form">Bilinear</a></li> <li><a href="/wiki/Sesquilinear_form" title="Sesquilinear form">Sesquilinear</a></li></ul></li> <li><a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">Norm</a></li> <li><a href="/wiki/Seminorm" title="Seminorm">Seminorm</a></li> <li><a href="/wiki/Sublinear_function" title="Sublinear function">Sublinear function</a></li> <li><a href="/wiki/Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of sets</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Absolutely_convex_set" title="Absolutely convex set">Absolutely convex/disk</a></li> <li><a href="/wiki/Absorbing_set" title="Absorbing set">Absorbing/Radial</a></li> <li><a href="/wiki/Affine_space" title="Affine space">Affine</a></li> <li><a href="/wiki/Balanced_set" title="Balanced set">Balanced/Circled</a></li> <li><a href="/wiki/Auxiliary_normed_space" title="Auxiliary normed space">Banach disks</a></li> <li><a href="/wiki/Bounding_point" title="Bounding point">Bounding points</a></li> <li><a href="/wiki/Bounded_set_(topological_vector_space)" title="Bounded set (topological vector space)">Bounded</a></li> <li><a href="/wiki/Complemented_subspace" title="Complemented subspace">Complemented subspace</a></li> <li><a href="/wiki/Convex_set" title="Convex set">Convex</a></li> <li><a href="/wiki/Convex_cone" title="Convex cone">Convex cone <span style="font-size:85%;">(subset)</span></a></li> <li><a href="/wiki/Cone_(linear_algebra)" class="mw-redirect" title="Cone (linear algebra)">Linear cone <span style="font-size:85%;">(subset)</span></a></li> <li><a href="/wiki/Extreme_point" title="Extreme point">Extreme point</a></li> <li><a href="/wiki/Totally_bounded_space#Topological_vector_spaces" title="Totally bounded space">Pre-compact/Totally bounded</a></li> <li><a href="/wiki/Prevalent_and_shy_sets" title="Prevalent and shy sets">Prevalent/Shy</a></li> <li><a href="/wiki/Radial_set" title="Radial set">Radial</a></li> <li><a href="/wiki/Star_domain" title="Star domain">Radially convex/Star-shaped</a></li> <li><a href="/wiki/Symmetric_set" title="Symmetric set">Symmetric</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set operations</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/Affine_hull" title="Affine hull">Affine hull</a></li> <li>(<a href="/wiki/Algebraic_interior#Relative_algebraic_interior" title="Algebraic interior">Relative</a>) <a href="/wiki/Algebraic_interior" title="Algebraic interior">Algebraic interior (core)</a></li> <li><a href="/wiki/Convex_hull" title="Convex hull">Convex hull</a></li> <li><a href="/wiki/Linear_span" title="Linear span">Linear span</a></li> <li><a href="/wiki/Minkowski_addition" title="Minkowski addition">Minkowski addition</a></li> <li><a href="/wiki/Polar_set" title="Polar set">Polar</a></li> <li>(<a href="/wiki/Algebraic_interior#Quasi_relative_interior" title="Algebraic interior">Quasi</a>) <a href="/wiki/Algebraic_interior#Relative_interior" title="Algebraic interior">Relative interior</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of TVSs</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/Asplund_space" title="Asplund space">Asplund</a></li> <li><a href="/wiki/Ptak_space" title="Ptak space">B-complete/Ptak</a></li> <li><a href="/wiki/Banach_space" title="Banach space">Banach</a></li> <li>(<a href="/wiki/Countably_barrelled_space" title="Countably barrelled space">Countably</a>) <a href="/wiki/Barrelled_space" title="Barrelled space">Barrelled</a></li> <li><a href="/wiki/BK-space" title="BK-space">BK-space</a></li> <li>(<a href="/wiki/Ultrabornological_space" title="Ultrabornological space">Ultra-</a>) <a href="/wiki/Bornological_space" title="Bornological space">Bornological</a></li> <li><a href="/wiki/Brauner_space" title="Brauner space">Brauner</a></li> <li><a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">Complete</a></li> <li><a href="/wiki/Convenient_vector_space" title="Convenient vector space">Convenient</a></li> <li><a href="/wiki/DF-space" title="DF-space">(DF)-space</a></li> <li><a href="/wiki/Distinguished_space" title="Distinguished space">Distinguished</a></li> <li><a href="/wiki/F-space" title="F-space">F-space</a></li> <li><a href="/wiki/FK-AK_space" title="FK-AK space">FK-AK space</a></li> <li><a href="/wiki/FK-space" title="FK-space">FK-space</a></li> <li><a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet</a> <ul><li><a href="/wiki/Differentiation_in_Fr%C3%A9chet_spaces#Tame_Fréchet_spaces" title="Differentiation in Fréchet spaces">tame Fréchet</a></li></ul></li> <li><a href="/wiki/Grothendieck_space" title="Grothendieck space">Grothendieck</a></li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert</a></li> <li><a href="/wiki/Infrabarreled_space" class="mw-redirect" title="Infrabarreled space">Infrabarreled</a></li> <li><a href="/wiki/Interpolation_space" title="Interpolation space">Interpolation space</a></li> <li><a href="/wiki/K-space_(functional_analysis)" title="K-space (functional analysis)">K-space</a></li> <li><a href="/wiki/LB-space" title="LB-space">LB-space</a></li> <li><a href="/wiki/LF-space" title="LF-space">LF-space</a></li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex space</a></li> <li><a href="/wiki/Mackey_space" title="Mackey space">Mackey</a></li> <li><a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">(Pseudo)Metrizable</a></li> <li><a href="/wiki/Montel_space" title="Montel space">Montel</a></li> <li><a href="/wiki/Quasibarrelled_space" class="mw-redirect" title="Quasibarrelled space">Quasibarrelled</a></li> <li><a href="/wiki/Quasi-complete" class="mw-redirect" title="Quasi-complete">Quasi-complete</a></li> <li><a href="/wiki/Quasinorm" title="Quasinorm">Quasinormed</a></li> <li>(<a href="/wiki/Polynomially_reflexive_space" title="Polynomially reflexive space">Polynomially</a></li> <li><a href="/wiki/Semi-reflexive_space" title="Semi-reflexive space">Semi-</a>) <a href="/wiki/Reflexive_space" title="Reflexive space">Reflexive</a></li> <li><a href="/wiki/Riesz_space" title="Riesz space">Riesz</a></li> <li><a href="/wiki/Schwartz_TVS" class="mw-redirect" title="Schwartz TVS">Schwartz</a></li> <li><a href="/wiki/Semi-complete" class="mw-redirect" title="Semi-complete">Semi-complete</a></li> <li><a href="/wiki/Smith_space" title="Smith space">Smith</a></li> <li><a href="/wiki/Stereotype_space" class="mw-redirect" title="Stereotype space">Stereotype</a></li> <li>(<a href="/wiki/B-convex_space" title="B-convex space">B</a></li> <li><a href="/wiki/Strictly_convex_space" title="Strictly convex space">Strictly</a></li> <li><a href="/wiki/Uniformly_convex_space" title="Uniformly convex space">Uniformly</a>) convex</li> <li>(<a href="/wiki/Quasi-ultrabarrelled_space" title="Quasi-ultrabarrelled space">Quasi-</a>) <a href="/wiki/Ultrabarrelled_space" title="Ultrabarrelled space">Ultrabarrelled</a></li> <li><a href="/wiki/Uniformly_smooth_space" title="Uniformly smooth space">Uniformly smooth</a></li> <li><a href="/wiki/Webbed_space" title="Webbed space">Webbed</a></li> 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