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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Manifold modeled on Banach spaces</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>Banach manifold</b> is a <a href="/wiki/Manifold" title="Manifold">manifold</a> modeled on <a href="/wiki/Banach_spaces" class="mw-redirect" title="Banach spaces">Banach spaces</a>. Thus it is a <a href="/wiki/Topological_space" title="Topological space">topological space</a> in which each point has a <a href="/wiki/Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">neighbourhood</a> <a href="/wiki/Homeomorphic" class="mw-redirect" title="Homeomorphic">homeomorphic</a> to an <a href="/wiki/Open_set" title="Open set">open set</a> in a Banach space (a more involved and formal definition is given below). Banach manifolds are one possibility of extending manifolds to <a href="/wiki/Infinity" title="Infinity">infinite</a> <a href="/wiki/Dimension" title="Dimension">dimensions</a>. </p><p>A further generalisation is to <a href="/wiki/Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifolds</a>, replacing Banach spaces by <a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet spaces</a>. On the other hand, a <a href="/wiki/Hilbert_manifold" title="Hilbert manifold">Hilbert manifold</a> is a special case of a Banach manifold in which the manifold is locally modeled on <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Banach_manifold&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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/Set_(mathematics)" title="Set (mathematics)">set</a>. An <b><a href="/wiki/Atlas_(topology)" title="Atlas (topology)">atlas</a> of class</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{r},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{r},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62edb90293494de273cf5db2341c768c38f5b6e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.418ex; height:2.676ex;" alt="{\displaystyle C^{r},}" /></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\geq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\geq 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/623bf4cb7b59cf1dc603b29b360034ae7982fd58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.956ex; height:2.509ex;" alt="{\displaystyle r\geq 0,}" /></span> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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 collection of pairs (called <b><a href="/wiki/Atlas_(topology)#Charts" title="Atlas (topology)">charts</a></b>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(U_{i},\varphi _{i}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(U_{i},\varphi _{i}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c58768131d44096e58524e9a06634d80b7772623" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.584ex; height:2.843ex;" alt="{\displaystyle \left(U_{i},\varphi _{i}\right),}" /></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in I,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in I,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42f9f22d39bd7568720b485fdb9ced8f99c1c63e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.462ex; height:2.509ex;" alt="{\displaystyle i\in I,}" /></span> such that </p> <ol><li>each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b21a6f475b0e68475c6019abe1fed0b415e0e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.387ex; height:2.509ex;" alt="{\displaystyle U_{i}}" /></span> is a <a href="/wiki/Subset" title="Subset">subset</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </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 the <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a> of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b21a6f475b0e68475c6019abe1fed0b415e0e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.387ex; height:2.509ex;" alt="{\displaystyle U_{i}}" /></span> is the whole 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> <li>each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70503774fb21be77396899900d3aa1e47d8f9e10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.32ex; height:2.176ex;" alt="{\displaystyle \varphi _{i}}" /></span> is a <a href="/wiki/Bijection" title="Bijection">bijection</a> from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b21a6f475b0e68475c6019abe1fed0b415e0e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.387ex; height:2.509ex;" alt="{\displaystyle U_{i}}" /></span> onto an <a href="/wiki/Open_subset" class="mw-redirect" title="Open subset">open subset</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 \varphi _{i}\left(U_{i}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{i}\left(U_{i}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11b1d718f80ca871a4161a60d74f4cb230f47019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.903ex; height:2.843ex;" alt="{\displaystyle \varphi _{i}\left(U_{i}\right)}" /></span> of some Banach space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/644a5eb6a688ef6e2751a89f914591096d7e16da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.162ex; height:2.509ex;" alt="{\displaystyle E_{i},}" /></span> and for any indices <span class="mwe-math-element"><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{\text{ and }}j,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>j</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i{\text{ and }}j,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc3cdf5f6493a73ccc8d0b47dca4b6cdfc573f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.316ex; height:2.509ex;" alt="{\displaystyle i{\text{ and }}j,}" /></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 \varphi _{i}\left(U_{i}\cap U_{j}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{i}\left(U_{i}\cap U_{j}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58d5700638644240b9d8c7a66edc00b5ae2c1849" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.983ex; height:3.009ex;" alt="{\displaystyle \varphi _{i}\left(U_{i}\cap U_{j}\right)}" /></span> is open 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 E_{i};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cac2e5eeef035085345162dbf5121394da0e7614" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.162ex; height:2.509ex;" alt="{\displaystyle E_{i};}" /></span></li> <li>the crossover map <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{j}\circ \varphi _{i}^{-1}:\varphi _{i}\left(U_{i}\cap U_{j}\right)\to \varphi _{j}\left(U_{i}\cap U_{j}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∘</mo> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>:</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">→</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{j}\circ \varphi _{i}^{-1}:\varphi _{i}\left(U_{i}\cap U_{j}\right)\to \varphi _{j}\left(U_{i}\cap U_{j}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a8eb4d35d9571891f8924047d4931cb4c82f49" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.105ex; height:3.343ex;" alt="{\displaystyle \varphi _{j}\circ \varphi _{i}^{-1}:\varphi _{i}\left(U_{i}\cap U_{j}\right)\to \varphi _{j}\left(U_{i}\cap U_{j}\right)}" /></span> is an <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth 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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /></span>-times continuously differentiable</a> function for every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j\in I;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>∈</mo> <mi>I</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i,j\in I;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f88c200cfea71ca280257b21933e3dab00bd35ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.454ex; height:2.509ex;" alt="{\displaystyle i,j\in I;}" /></span> that is, the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /></span>th <a href="/wiki/Fr%C3%A9chet_derivative" title="Fréchet derivative">Fréchet derivative</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} ^{r}\left(\varphi _{j}\circ \varphi _{i}^{-1}\right):\varphi _{i}\left(U_{i}\cap U_{j}\right)\to \mathrm {Lin} \left(E_{i}^{r};E_{j}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∘</mo> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>:</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <mo>;</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} ^{r}\left(\varphi _{j}\circ \varphi _{i}^{-1}\right):\varphi _{i}\left(U_{i}\cap U_{j}\right)\to \mathrm {Lin} \left(E_{i}^{r};E_{j}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b64542dc1ead3f3070463c5561dd91628245969" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.131ex; height:3.343ex;" alt="{\displaystyle \mathrm {d} ^{r}\left(\varphi _{j}\circ \varphi _{i}^{-1}\right):\varphi _{i}\left(U_{i}\cap U_{j}\right)\to \mathrm {Lin} \left(E_{i}^{r};E_{j}\right)}" /></span> exists and is a <a href="/wiki/Continuous_function" title="Continuous function">continuous function</a> with respect to the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba9f6e3041b052cf13a0ede4ecf35fb4c9cd16c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.515ex; height:2.509ex;" alt="{\displaystyle E_{i}}" /></span>-<a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a> <a href="/wiki/Topology" title="Topology">topology</a> on subsets of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba9f6e3041b052cf13a0ede4ecf35fb4c9cd16c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.515ex; height:2.509ex;" alt="{\displaystyle E_{i}}" /></span> and the <a href="/wiki/Operator_norm" title="Operator norm">operator norm</a> topology on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Lin} \left(E_{i}^{r};E_{j}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Lin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <mo>;</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Lin} \left(E_{i}^{r};E_{j}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7600907090bd6acefbfdef942719edff36c70360" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.595ex; height:3.176ex;" alt="{\displaystyle \operatorname {Lin} \left(E_{i}^{r};E_{j}\right).}" /></span></li></ol> <p>One can then show that there is a unique <a href="/wiki/Topology" title="Topology">topology</a> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> such that each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b21a6f475b0e68475c6019abe1fed0b415e0e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.387ex; height:2.509ex;" alt="{\displaystyle U_{i}}" /></span> is open and each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70503774fb21be77396899900d3aa1e47d8f9e10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.32ex; height:2.176ex;" alt="{\displaystyle \varphi _{i}}" /></span> is a <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a>. Very often, this topological space is assumed to be a <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff space</a>, but this is not necessary from the point of view of the formal definition. </p><p>If all the Banach spaces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba9f6e3041b052cf13a0ede4ecf35fb4c9cd16c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.515ex; height:2.509ex;" alt="{\displaystyle E_{i}}" /></span> are equal to the same space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89862747e88ca143e979241a9a243b5ef66ddc67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.422ex; height:2.509ex;" alt="{\displaystyle E,}" /></span> the atlas is called an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" /></span><b>-atlas</b>. However, it is not <i><a href="https://en.wiktionary.org/wiki/A_priori" class="extiw" title="wiktionary:A priori">a priori</a></i> necessary that the Banach spaces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba9f6e3041b052cf13a0ede4ecf35fb4c9cd16c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.515ex; height:2.509ex;" alt="{\displaystyle E_{i}}" /></span> be the same space, or even <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> as <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector spaces</a>. However, if two charts <span class="mwe-math-element"><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(U_{i},\varphi _{i}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(U_{i},\varphi _{i}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f97dfe45d716bc3eddc2ef2b79b97e000c2203d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.55ex; height:2.843ex;" alt="{\displaystyle \left(U_{i},\varphi _{i}\right)}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(U_{j},\varphi _{j}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(U_{j},\varphi _{j}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61c98ec367b11b02a20d718829797151d1da8d87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.77ex; height:3.009ex;" alt="{\displaystyle \left(U_{j},\varphi _{j}\right)}" /></span> are 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 U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b21a6f475b0e68475c6019abe1fed0b415e0e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.387ex; height:2.509ex;" alt="{\displaystyle U_{i}}" /></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 U_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d85201a34558a919f8cc0e193492932eedfc6ce7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.497ex; height:2.843ex;" alt="{\displaystyle U_{j}}" /></span> have a non-empty <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a>, a quick examination of the <a href="/wiki/Derivative_(generalizations)" class="mw-redirect" title="Derivative (generalizations)">derivative</a> of the crossover map <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{j}\circ \varphi _{i}^{-1}:\varphi _{i}\left(U_{i}\cap U_{j}\right)\to \varphi _{j}\left(U_{i}\cap U_{j}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∘</mo> <msubsup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>:</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">→</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{j}\circ \varphi _{i}^{-1}:\varphi _{i}\left(U_{i}\cap U_{j}\right)\to \varphi _{j}\left(U_{i}\cap U_{j}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a8eb4d35d9571891f8924047d4931cb4c82f49" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.105ex; height:3.343ex;" alt="{\displaystyle \varphi _{j}\circ \varphi _{i}^{-1}:\varphi _{i}\left(U_{i}\cap U_{j}\right)\to \varphi _{j}\left(U_{i}\cap U_{j}\right)}" /></span> shows that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba9f6e3041b052cf13a0ede4ecf35fb4c9cd16c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.515ex; height:2.509ex;" alt="{\displaystyle E_{i}}" /></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 E_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a01babf2293d37d9ff73b61202cfc3a3933eaf19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.625ex; height:2.843ex;" alt="{\displaystyle E_{j}}" /></span> must indeed be isomorphic as topological vector spaces. Furthermore, the set of points <span class="mwe-math-element"><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> for which there is a chart <span class="mwe-math-element"><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(U_{i},\varphi _{i}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(U_{i},\varphi _{i}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f97dfe45d716bc3eddc2ef2b79b97e000c2203d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.55ex; height:2.843ex;" alt="{\displaystyle \left(U_{i},\varphi _{i}\right)}" /></span> 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/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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b21a6f475b0e68475c6019abe1fed0b415e0e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.387ex; height:2.509ex;" alt="{\displaystyle U_{i}}" /></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 E_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba9f6e3041b052cf13a0ede4ecf35fb4c9cd16c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.515ex; height:2.509ex;" alt="{\displaystyle E_{i}}" /></span> isomorphic to a given Banach space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" /></span> is both open and <a href="/wiki/Closed_subset" class="mw-redirect" title="Closed subset">closed</a>. Hence, one can without loss of generality assume that, on each <a href="/wiki/Connected_space" title="Connected space">connected component</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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> the atlas is an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" /></span>-atlas for some fixed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a2566d01f104ef084ea424b8b35c2534f7f902b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.422ex; height:2.176ex;" alt="{\displaystyle E.}" /></span> </p><p>A new chart <span class="mwe-math-element"><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,\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>U</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (U,\varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8223ff975a26d1266faaa46de9b88f2e469a15d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.146ex; height:2.843ex;" alt="{\displaystyle (U,\varphi )}" /></span> is called <b>compatible</b> with a given atlas <span class="mwe-math-element"><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\{\left(U_{i},\varphi _{i}\right):i\in I\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>:</mo> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{\left(U_{i},\varphi _{i}\right):i\in I\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3516159d69ce8bc26c1ccc3e6299fd40876a3307" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.627ex; height:2.843ex;" alt="{\displaystyle \left\{\left(U_{i},\varphi _{i}\right):i\in I\right\}}" /></span> if the crossover map <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{i}\circ \varphi ^{-1}:\varphi \left(U\cap U_{i}\right)\to \varphi _{i}\left(U\cap U_{i}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∘</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>:</mo> <mi>φ</mi> <mrow> <mo>(</mo> <mrow> <mi>U</mi> <mo>∩</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">→</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>U</mi> <mo>∩</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{i}\circ \varphi ^{-1}:\varphi \left(U\cap U_{i}\right)\to \varphi _{i}\left(U\cap U_{i}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2580da44a60012bc0ddea7b46647349e86522d29" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.656ex; height:3.176ex;" alt="{\displaystyle \varphi _{i}\circ \varphi ^{-1}:\varphi \left(U\cap U_{i}\right)\to \varphi _{i}\left(U\cap U_{i}\right)}" /></span> is an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /></span>-times continuously differentiable function for every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in I.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in I.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3627df37846ed8181157cbb3195735ec0988baa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.462ex; height:2.176ex;" alt="{\displaystyle i\in I.}" /></span> Two atlases are called compatible if every chart in one is compatible with the other atlas. Compatibility defines an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> on the class of all possible atlases on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> </p><p>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 C^{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdefeb0e26e08fdf32d93d61a5382c800515ddf2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.772ex; height:2.343ex;" alt="{\displaystyle C^{r}}" /></span><b>-manifold</b> structure on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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 then defined to be a choice of equivalence class of atlases on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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 class <span class="mwe-math-element"><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^{r}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{r}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11819f2271f4ebb93ce39536e760d6939a038774" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.418ex; height:2.343ex;" alt="{\displaystyle C^{r}.}" /></span> If all the Banach spaces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba9f6e3041b052cf13a0ede4ecf35fb4c9cd16c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.515ex; height:2.509ex;" alt="{\displaystyle E_{i}}" /></span> are isomorphic as topological vector spaces (which is guaranteed to be the case if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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 href="/wiki/Connected_space" title="Connected space">connected</a>), then an equivalent atlas can be found for which they are all equal to some Banach space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a2566d01f104ef084ea424b8b35c2534f7f902b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.422ex; height:2.176ex;" alt="{\displaystyle E.}" /></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}"> <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 then called an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" /></span><b>-manifold</b>, or one says 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 <b>modeled</b> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a2566d01f104ef084ea424b8b35c2534f7f902b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.422ex; height:2.176ex;" alt="{\displaystyle E.}" /></span> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Banach_manifold&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every Banach space can be canonically identified as a Banach manifold. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\|\,\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> is a Banach space, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <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 Banach manifold with an atlas containing a single, globally-defined chart (the <a href="/wiki/Identity_function" title="Identity function">identity map</a>). </p><p>Similarly, 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 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> is an open subset of some Banach space 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 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> is a Banach manifold. (See the <a href="/wiki/Classification_theorem" title="Classification theorem">classification theorem</a> below.) </p> <div class="mw-heading mw-heading2"><h2 id="Classification_up_to_homeomorphism">Classification up to homeomorphism</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Banach_manifold&action=edit&section=3" title="Edit section: Classification up to homeomorphism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is by no means true that a finite-dimensional manifold of dimension <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> is <em>globally</em> homeomorphic 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 \mathbb {R} ^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7035fcb9fe3ebecc6bc9f372f82d0352202c8bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{n},}" /></span> or even an 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 \mathbb {R} ^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76ef548febfc9981762740107858be9e3a5576c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.543ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}.}" /></span> However, in an infinite-dimensional setting, it is possible to classify "<a href="/wiki/Well-behaved" class="mw-redirect" title="Well-behaved">well-behaved</a>" Banach manifolds up to homeomorphism quite nicely. A 1969 theorem of <a href="/wiki/David_W._Henderson" title="David W. Henderson">David Henderson</a><sup id="cite_ref-FOOTNOTEHenderson1969_1-0" class="reference"><a href="#cite_note-FOOTNOTEHenderson1969-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> states that every infinite-dimensional, <a href="/wiki/Separable_space" title="Separable space">separable</a>, <a href="/wiki/Metric_space" title="Metric space">metric</a> Banach manifold <span class="mwe-math-element"><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> can be <a href="/wiki/Embedding" title="Embedding">embedded</a> as an open subset of the infinite-dimensional, separable Hilbert space, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" /></span> (up to linear isomorphism, there is only one such space, usually identified with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91f1f909abd70bd3d8fff0f7ae1ac23052387e18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{2}}" /></span>). In fact, Henderson's result is stronger: the same conclusion holds for any metric manifold modeled on a separable infinite-dimensional <a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet space</a>. </p><p>The embedding homeomorphism can be used as a global chart for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}"> <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, in the infinite-dimensional, separable, metric case, the "only" Banach manifolds are the open subsets of Hilbert space. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Banach_manifold&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Banach_bundle" title="Banach bundle">Banach bundle</a> – vector bundle whose fibres form Banach spaces<span style="display:none" class="category-wikidata-fallback-annotation">Pages displaying wikidata descriptions as a fallback</span></li> <li><a href="/wiki/Differentiation_in_Fr%C3%A9chet_spaces" title="Differentiation in Fréchet spaces">Differentiation in Fréchet spaces</a></li> <li><a href="/wiki/Finsler_manifold" title="Finsler manifold">Finsler manifold</a> – Generalization of Riemannian manifolds</li> <li><a href="/wiki/Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a> – topological space modeled on a Fréchet space in much the same way as a manifold is modeled on a Euclidean space<span style="display:none" class="category-wikidata-fallback-annotation">Pages displaying wikidata descriptions as a fallback</span></li> <li><a href="/wiki/Global_analysis" title="Global analysis">Global analysis</a> – which uses Banach manifolds and other kinds of infinite-dimensional manifolds</li> <li><a href="/wiki/Hilbert_manifold" title="Hilbert manifold">Hilbert manifold</a> – Manifold modelled on Hilbert spaces</li></ul> <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=Banach_manifold&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-FOOTNOTEHenderson1969-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHenderson1969_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHenderson1969">Henderson 1969</a>.</span> </li> </ol></div></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist"> </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="CITEREFAbrahamMarsdenRatiu1988" class="citation book cs1">Abraham, Ralph; Marsden, J. E.; Ratiu, Tudor (1988). <i>Manifolds, Tensor Analysis, and Applications</i>. New York: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-96790-7" title="Special:BookSources/0-387-96790-7"><bdi>0-387-96790-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Manifolds%2C+Tensor+Analysis%2C+and+Applications&rft.place=New+York&rft.pub=Springer&rft.date=1988&rft.isbn=0-387-96790-7&rft.aulast=Abraham&rft.aufirst=Ralph&rft.au=Marsden%2C+J.+E.&rft.au=Ratiu%2C+Tudor&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABanach+manifold" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAnderson1969" class="citation journal cs1">Anderson, R. D. (1969). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/proc/1969-023-03/S0002-9939-1969-0248883-5/S0002-9939-1969-0248883-5.pdf">"Strongly negligible sets in Fréchet manifolds"</a> <span class="cs1-format">(PDF)</span>. <i>Bulletin of the American Mathematical Society</i>. <b>75</b> (1). 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D.; Schori, R. (1969). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/tran/1969-142-00/S0002-9947-1969-0246327-5/S0002-9947-1969-0246327-5.pdf">"Factors of infinite-dimensional manifolds"</a> <span class="cs1-format">(PDF)</span>. <i>Transactions of the American Mathematical Society</i>. <b>142</b>. American Mathematical Society (AMS): <span class="nowrap">315–</span>330. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9947-1969-0246327-5">10.1090/s0002-9947-1969-0246327-5</a>. <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/0002-9947">0002-9947</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=Factors+of+infinite-dimensional+manifolds&rft.volume=142&rft.pages=%3Cspan+class%3D%22nowrap%22%3E315-%3C%2Fspan%3E330&rft.date=1969&rft_id=info%3Adoi%2F10.1090%2Fs0002-9947-1969-0246327-5&rft.issn=0002-9947&rft.aulast=Anderson&rft.aufirst=R.+D.&rft.au=Schori%2C+R.&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Ftran%2F1969-142-00%2FS0002-9947-1969-0246327-5%2FS0002-9947-1969-0246327-5.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABanach+manifold" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHenderson1969" class="citation journal cs1">Henderson, David W. (1969). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1969-12276-7">"Infinite-dimensional manifolds are open subsets of Hilbert space"</a>. <i>Bull. Amer. Math. Soc</i>. <b>75</b> (4): <span class="nowrap">759–</span>762. <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%2FS0002-9904-1969-12276-7">10.1090/S0002-9904-1969-12276-7</a></span>. <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=0247634">0247634</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bull.+Amer.+Math.+Soc.&rft.atitle=Infinite-dimensional+manifolds+are+open+subsets+of+Hilbert+space&rft.volume=75&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E759-%3C%2Fspan%3E762&rft.date=1969&rft_id=info%3Adoi%2F10.1090%2FS0002-9904-1969-12276-7&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0247634%23id-name%3DMR&rft.aulast=Henderson&rft.aufirst=David+W.&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252FS0002-9904-1969-12276-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABanach+manifold" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLang1972" class="citation book cs1"><a href="/wiki/Serge_Lang" title="Serge Lang">Lang, Serge</a> (1972). <i>Differential manifolds</i>. Reading, Mass.–London–Don Mills, Ont.: Addison-Wesley Publishing Co., Inc.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+manifolds&rft.place=Reading%2C+Mass.%E2%80%93London%E2%80%93Don+Mills%2C+Ont.&rft.pub=Addison-Wesley+Publishing+Co.%2C+Inc.&rft.date=1972&rft.aulast=Lang&rft.aufirst=Serge&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABanach+manifold" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFZeidler1997" class="citation book cs1">Zeidler, Eberhard (1997). <i>Nonlinear functional analysis and its Applications. Vol.4</i>. Springer-Verlag New York Inc.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Nonlinear+functional+analysis+and+its+Applications.+Vol.4&rft.pub=Springer-Verlag+New+York+Inc.&rft.date=1997&rft.aulast=Zeidler&rft.aufirst=Eberhard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABanach+manifold" 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 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.navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Manifolds" title="Template:Manifolds"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Manifolds" title="Template talk:Manifolds"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Manifolds" title="Special:EditPage/Template:Manifolds"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Manifolds_(Glossary,_List,_Category)273" style="font-size:114%;margin:0 4em"><a href="/wiki/Manifold" title="Manifold">Manifolds</a> (<a href="/wiki/Glossary_of_differential_geometry_and_topology" title="Glossary of differential geometry and topology">Glossary</a>, <a href="/wiki/List_of_manifolds" title="List of manifolds">List</a>, <a href="/wiki/Category:Manifolds" title="Category:Manifolds">Category</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/Topological_manifold" title="Topological manifold">Topological manifold</a> <ul><li><a href="/wiki/Atlas_(topology)" title="Atlas (topology)">Atlas</a></li></ul></li> <li><a href="/wiki/Differentiable_manifold" title="Differentiable manifold">Differentiable/Smooth manifold</a> <ul><li><a href="/wiki/Differential_structure" title="Differential structure">Differential structure</a></li> <li><a href="/wiki/Smooth_structure" title="Smooth structure">Smooth atlas</a></li></ul></li> <li><a href="/wiki/Submanifold" title="Submanifold">Submanifold</a></li> <li><a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></li> <li><a href="/wiki/Smoothness" title="Smoothness">Smooth map</a></li> <li><a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li> <li><a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li> <li><a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a></li> <li><a href="/wiki/Vector_field" title="Vector field">Vector field</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results <span style="font-size:85%;"><a href="/wiki/Category:Theorems_in_differential_geometry" title="Category:Theorems in differential geometry">(list)</a></span></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/Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index</a></li> <li><a href="/wiki/Darboux%27s_theorem" title="Darboux's theorem">Darboux's</a></li> <li><a href="/wiki/De_Rham_cohomology#De_Rham's_theorem" title="De Rham cohomology">De Rham's</a></li> <li><a href="/wiki/Frobenius_theorem_(differential_topology)" title="Frobenius theorem (differential topology)">Frobenius</a></li> <li><a href="/wiki/Generalized_Stokes_theorem" title="Generalized Stokes theorem">Generalized Stokes</a></li> <li><a href="/wiki/Hopf%E2%80%93Rinow_theorem" title="Hopf–Rinow theorem">Hopf–Rinow</a></li> <li><a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's</a></li> <li><a href="/wiki/Sard%27s_theorem" title="Sard's theorem">Sard's</a></li> <li><a href="/wiki/Whitney_embedding_theorem" title="Whitney embedding theorem">Whitney embedding</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Smoothness" title="Smoothness">Maps</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/Differentiable_curve" title="Differentiable curve">Curve</a></li> <li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a> <ul><li><a href="/wiki/Local_diffeomorphism" title="Local diffeomorphism">Local</a></li></ul></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Exponential_map_(Riemannian_geometry)" title="Exponential map (Riemannian geometry)">Exponential map</a> <ul><li><a href="/wiki/Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">in Lie theory</a></li></ul></li> <li><a href="/wiki/Foliation" title="Foliation">Foliation</a></li> <li><a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">Immersion</a></li> <li><a href="/wiki/Integral_curve" title="Integral curve">Integral curve</a></li> <li><a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a></li> <li><a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">Section</a></li> <li><a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of<br />manifolds</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/Closed_manifold" title="Closed manifold">Closed</a></li> <li><a href="/wiki/Collapsing_manifold" title="Collapsing manifold">Collapsing</a></li> <li><a href="/wiki/Complete_manifold" title="Complete manifold">Complete</a></li> <li>(<a href="/wiki/Almost_complex_manifold" title="Almost complex manifold">Almost</a>) <a href="/wiki/Complex_manifold" title="Complex manifold">Complex</a></li> <li>(<a href="/wiki/Almost-contact_manifold" title="Almost-contact manifold">Almost</a>) <a href="/wiki/Contact_manifold" class="mw-redirect" title="Contact manifold">Contact</a></li> <li><a href="/wiki/Fibered_manifold" title="Fibered manifold">Fibered</a></li> <li><a href="/wiki/Finsler_manifold" title="Finsler manifold">Finsler</a></li> <li>(<a href="/wiki/Almost_flat_manifold" title="Almost flat manifold">Almost</a>) <a href="/wiki/Flat_manifold" title="Flat manifold">Flat</a></li> <li><a href="/wiki/G-structure_on_a_manifold" title="G-structure on a manifold">G-structure</a></li> <li><a href="/wiki/Hadamard_manifold" title="Hadamard manifold">Hadamard</a></li> <li><a href="/wiki/Hermitian_manifold" title="Hermitian manifold">Hermitian</a></li> <li><a href="/wiki/Hyperbolic_manifold" title="Hyperbolic manifold">Hyperbolic</a></li> <li><a href="/wiki/K%C3%A4hler_manifold" title="Kähler manifold">Kähler</a></li> <li><a href="/wiki/Kenmotsu_manifold" title="Kenmotsu manifold">Kenmotsu</a></li> <li><a href="/wiki/Lie_group" title="Lie group">Lie group</a> <ul><li><a href="/wiki/Lie_group%E2%80%93Lie_algebra_correspondence" title="Lie group–Lie algebra correspondence">Lie algebra</a></li></ul></li> <li><a href="/wiki/Manifold_with_boundary" class="mw-redirect" title="Manifold with boundary">Manifold with boundary</a></li> <li><a href="/wiki/Nilmanifold" title="Nilmanifold">Nilmanifold</a></li> <li><a href="/wiki/Orientability" title="Orientability">Oriented</a></li> <li><a href="/wiki/Parallelizable_manifold" title="Parallelizable manifold">Parallelizable</a></li> <li><a href="/wiki/Poisson_manifold" title="Poisson manifold">Poisson</a></li> <li><a href="/wiki/Prime_manifold" title="Prime manifold">Prime</a></li> <li><a href="/wiki/Quaternionic_manifold" title="Quaternionic manifold">Quaternionic</a></li> <li><a href="/wiki/Hypercomplex_manifold" title="Hypercomplex manifold">Hypercomplex</a></li> <li>(<a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo−</a>, <a href="/wiki/Sub-Riemannian_manifold" title="Sub-Riemannian manifold">Sub−</a>) <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian</a></li> <li><a href="/wiki/Rizza_manifold" title="Rizza manifold">Rizza</a></li> <li>(<a href="/wiki/Almost_symplectic_manifold" title="Almost symplectic manifold">Almost</a>) <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">Symplectic</a></li> <li><a href="/wiki/Tame_manifold" title="Tame manifold">Tame</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Tensor" title="Tensor">Tensors</a></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><th scope="row" class="navbox-group" style="width:1%">Vectors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Distribution_(differential_geometry)" title="Distribution (differential geometry)">Distribution</a></li> <li><a href="/wiki/Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket</a></li> <li><a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li> <li><a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a> <ul><li><a href="/wiki/Tangent_bundle" title="Tangent bundle">bundle</a></li></ul></li> <li><a href="/wiki/Torsion_tensor" title="Torsion tensor">Torsion</a></li> <li><a href="/wiki/Vector_field" title="Vector field">Vector field</a></li> <li><a href="/wiki/Vector_flow" title="Vector flow">Vector flow</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Covectors</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/Closed_and_exact_differential_forms" title="Closed and exact differential forms">Closed/Exact</a></li> <li><a href="/wiki/Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li> <li><a href="/wiki/Cotangent_space" title="Cotangent space">Cotangent space</a> <ul><li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">bundle</a></li></ul></li> <li><a href="/wiki/De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a> <ul><li><a href="/wiki/Vector-valued_differential_form" title="Vector-valued differential form">Vector-valued</a></li></ul></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Interior_product" title="Interior product">Interior product</a></li> <li><a href="/wiki/Pullback_(differential_geometry)" title="Pullback (differential geometry)">Pullback</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a> <ul><li><a href="/wiki/Ricci_flow" title="Ricci flow">flow</a></li></ul></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Tensor_field" title="Tensor field">Tensor field</a> <ul><li><a href="/wiki/Tensor_density" title="Tensor density">density</a></li></ul></li> <li><a href="/wiki/Volume_form" title="Volume form">Volume form</a></li> <li><a href="/wiki/Wedge_product" class="mw-redirect" title="Wedge product">Wedge product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Fiber_bundle" title="Fiber bundle">Bundles</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/Adjoint_bundle" title="Adjoint bundle">Adjoint</a></li> <li><a href="/wiki/Affine_bundle" title="Affine bundle">Affine</a></li> <li><a href="/wiki/Associated_bundle" title="Associated bundle">Associated</a></li> <li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">Cotangent</a></li> <li><a href="/wiki/Dual_bundle" title="Dual bundle">Dual</a></li> <li><a href="/wiki/Fiber_bundle" title="Fiber bundle">Fiber</a></li> <li>(<a href="/wiki/Cofibration" title="Cofibration">Co</a>) <a href="/wiki/Fibration" title="Fibration">Fibration</a></li> <li><a href="/wiki/Jet_bundle" title="Jet bundle">Jet</a></li> <li><a href="/wiki/Lie_algebra_bundle" title="Lie algebra bundle">Lie algebra</a></li> <li>(<a href="/wiki/Stable_normal_bundle" title="Stable normal bundle">Stable</a>) <a href="/wiki/Normal_bundle" title="Normal bundle">Normal</a></li> <li><a href="/wiki/Principal_bundle" title="Principal bundle">Principal</a></li> <li><a href="/wiki/Spinor_bundle" title="Spinor bundle">Spinor</a></li> <li><a href="/wiki/Subbundle" title="Subbundle">Subbundle</a></li> <li><a href="/wiki/Tangent_bundle" title="Tangent bundle">Tangent</a></li> <li><a href="/wiki/Tensor_bundle" title="Tensor bundle">Tensor</a></li> <li><a href="/wiki/Vector_bundle" title="Vector bundle">Vector</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Connection_(mathematics)" title="Connection (mathematics)">Connections</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/Affine_connection" title="Affine connection">Affine</a></li> <li><a href="/wiki/Cartan_connection" title="Cartan connection">Cartan</a></li> <li><a href="/wiki/Ehresmann_connection" title="Ehresmann connection">Ehresmann</a></li> <li><a href="/wiki/Connection_form" title="Connection form">Form</a></li> <li><a href="/wiki/Connection_(fibred_manifold)" title="Connection (fibred manifold)">Generalized</a></li> <li><a href="/wiki/Koszul_connection" class="mw-redirect" title="Koszul connection">Koszul</a></li> <li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita</a></li> <li><a href="/wiki/Connection_(principal_bundle)" title="Connection (principal bundle)">Principal</a></li> <li><a href="/wiki/Connection_(vector_bundle)" title="Connection (vector bundle)">Vector</a></li> <li><a href="/wiki/Parallel_transport" title="Parallel transport">Parallel transport</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classification_of_manifolds" title="Classification of manifolds">Classification of manifolds</a></li> <li><a href="/wiki/Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li> <li><a href="/wiki/History_of_manifolds_and_varieties" title="History of manifolds and varieties">History</a></li> <li><a href="/wiki/Morse_theory" title="Morse theory">Morse theory</a></li> <li><a href="/wiki/Moving_frame" title="Moving frame">Moving frame</a></li> <li><a href="/wiki/Singularity_theory" title="Singularity theory">Singularity theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</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 class="mw-selflink selflink">Banach manifold</a></li> <li><a href="/wiki/Diffeology" title="Diffeology">Diffeology</a></li> <li><a href="/wiki/Diffiety" title="Diffiety">Diffiety</a></li> <li><a href="/wiki/Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a></li> <li><a href="/wiki/K-theory" title="K-theory">K-theory</a></li> <li><a href="/wiki/Orbifold" title="Orbifold">Orbifold</a></li> <li><a href="/wiki/Secondary_calculus_and_cohomological_physics" title="Secondary calculus and cohomological physics">Secondary calculus</a> <ul><li><a href="/wiki/Differential_calculus_over_commutative_algebras" title="Differential calculus over commutative algebras">over commutative algebras</a></li></ul></li> <li><a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">Sheaf</a></li> <li><a href="/wiki/Stratifold" title="Stratifold">Stratifold</a></li> <li><a href="/wiki/Supermanifold" title="Supermanifold">Supermanifold</a></li> <li><a href="/wiki/Stratified_space" title="Stratified space">Stratified space</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="Analysis_in_topological_vector_spaces159" 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:Analysis_in_topological_vector_spaces" title="Template:Analysis in topological vector spaces"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Analysis_in_topological_vector_spaces" title="Template talk:Analysis in topological vector spaces"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Analysis_in_topological_vector_spaces" title="Special:EditPage/Template:Analysis in topological vector spaces"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Analysis_in_topological_vector_spaces159" style="font-size:114%;margin:0 4em"><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</a> in <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector spaces</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_Wiener_space" title="Abstract Wiener space">Abstract Wiener space</a> <ul><li><a href="/wiki/Classical_Wiener_space" title="Classical Wiener space">Classical Wiener space</a></li></ul></li> <li><a href="/wiki/Bochner_space" title="Bochner space">Bochner space</a></li> <li><a href="/wiki/Convex_series" title="Convex series">Convex series</a></li> <li><a href="/wiki/Cylinder_set_measure" title="Cylinder set measure">Cylinder set measure</a></li> <li><a href="/wiki/Infinite-dimensional_vector_function" title="Infinite-dimensional vector function">Infinite-dimensional vector function</a></li> <li><a href="/wiki/Matrix_calculus" title="Matrix calculus">Matrix calculus</a></li> <li><a href="/wiki/Vector_calculus" title="Vector calculus">Vector calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Derivative" title="Derivative">Derivatives</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/Differentiable_vector%E2%80%93valued_functions_from_Euclidean_space" title="Differentiable vector–valued functions from Euclidean space">Differentiable vector–valued functions from Euclidean space</a></li> <li><a href="/wiki/Differentiation_in_Fr%C3%A9chet_spaces" title="Differentiation in Fréchet spaces">Differentiation in Fréchet spaces</a></li> <li><a href="/wiki/Fr%C3%A9chet_derivative" title="Fréchet derivative">Fréchet derivative</a> <ul><li><a href="/wiki/Total_derivative" title="Total derivative">Total</a></li></ul></li> <li><a href="/wiki/Functional_derivative" title="Functional derivative">Functional derivative</a></li> <li><a href="/wiki/Gateaux_derivative" title="Gateaux derivative">Gateaux derivative</a> <ul><li><a href="/wiki/Directional_derivative" title="Directional derivative">Directional</a></li></ul></li> <li><a href="/wiki/Generalizations_of_the_derivative" title="Generalizations of the derivative">Generalizations of the derivative</a></li> <li><a href="/wiki/Hadamard_derivative" title="Hadamard derivative">Hadamard derivative</a></li> <li><a href="/wiki/Infinite-dimensional_holomorphy" title="Infinite-dimensional holomorphy">Holomorphic</a></li> <li><a href="/wiki/Quasi-derivative" title="Quasi-derivative">Quasi-derivative</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Measurability</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/Besov_measure" title="Besov measure">Besov measure</a></li> <li><a href="/wiki/Cylinder_set_measure" title="Cylinder set measure">Cylinder set measure</a> <ul><li><a href="/wiki/Canonical_Gaussian_cylinder_set_measure" class="mw-redirect" title="Canonical Gaussian cylinder set measure">Canonical Gaussian</a></li> <li><a href="/wiki/Classical_Wiener_measure" class="mw-redirect" title="Classical Wiener measure">Classical Wiener measure</a></li></ul></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measure</a> like <a href="/wiki/Set_function" title="Set function">set functions</a> <ul><li><a href="/wiki/Gaussian_measure#Infinite-dimensional_spaces" title="Gaussian measure">infinite-dimensional Gaussian measure</a></li> <li><a href="/wiki/Projection-valued_measure" title="Projection-valued measure">Projection-valued</a></li> <li><a href="/wiki/Vector_measure" title="Vector measure">Vector</a></li></ul></li> <li><a href="/wiki/Bochner_measurable_function" title="Bochner measurable function">Bochner</a> / <a href="/wiki/Weakly_measurable_function" title="Weakly measurable function">Weakly</a> / <a href="/wiki/Strongly_measurable_function" title="Strongly measurable function">Strongly</a> <a href="/wiki/Measurable_function" title="Measurable function">measurable function</a></li> <li><a href="/wiki/Radonifying_function" title="Radonifying function">Radonifying function</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Integral" title="Integral">Integrals</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/Bochner_integral" title="Bochner integral">Bochner</a></li> <li><a href="/wiki/Direct_integral" title="Direct integral">Direct integral</a></li> <li><a href="/wiki/Dunford_integral" class="mw-redirect" title="Dunford integral">Dunford</a></li> <li><a href="/wiki/Pettis_integral" title="Pettis integral">Gelfand–Pettis/Weak</a></li> <li><a href="/wiki/Regulated_integral" title="Regulated integral">Regulated</a></li> <li><a href="/wiki/Paley%E2%80%93Wiener_integral" title="Paley–Wiener integral">Paley–Wiener</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Results</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cameron%E2%80%93Martin_theorem" title="Cameron–Martin theorem">Cameron–Martin theorem</a></li> <li><a href="/wiki/Inverse_function_theorem" title="Inverse function theorem">Inverse function theorem</a> <ul><li><a href="/wiki/Nash%E2%80%93Moser_theorem" title="Nash–Moser theorem">Nash–Moser theorem</a></li></ul></li> <li><a href="/wiki/Feldman%E2%80%93H%C3%A1jek_theorem" title="Feldman–Hájek theorem">Feldman–Hájek theorem</a></li> <li><a href="/wiki/Infinite-dimensional_Lebesgue_measure" title="Infinite-dimensional Lebesgue measure">No infinite-dimensional Lebesgue measure</a></li> <li><a href="/wiki/Sazonov%27s_theorem" title="Sazonov's theorem">Sazonov's theorem</a></li> <li><a href="/wiki/Structure_theorem_for_Gaussian_measures" title="Structure theorem for Gaussian measures">Structure theorem for Gaussian measures</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">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/Crinkled_arc" title="Crinkled arc">Crinkled arc</a></li> <li><a href="/wiki/Covariance_operator" title="Covariance operator">Covariance operator</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Functional_calculus" title="Functional calculus">Functional calculus</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/Borel_functional_calculus" title="Borel functional calculus">Borel functional calculus</a></li> <li><a href="/wiki/Continuous_functional_calculus" title="Continuous functional calculus">Continuous functional calculus</a></li> <li><a href="/wiki/Holomorphic_functional_calculus" title="Holomorphic functional calculus">Holomorphic functional calculus</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-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Banach manifold</a> (<a href="/wiki/Banach_bundle" title="Banach bundle">bundle</a>)</li> <li><a href="/wiki/Convenient_vector_space" title="Convenient vector space">Convenient vector space</a></li> <li><a href="/wiki/Choquet_theory" title="Choquet theory">Choquet theory</a></li> <li><a href="/wiki/Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a></li> <li><a href="/wiki/Hilbert_manifold" title="Hilbert manifold">Hilbert manifold</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /></div><div role="navigation" class="navbox" aria-labelledby="Banach_space_topics288" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231" /><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Banach_spaces" title="Template:Banach spaces"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Banach_spaces" title="Template talk:Banach spaces"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Banach_spaces" title="Special:EditPage/Template:Banach spaces"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Banach_space_topics288" style="font-size:114%;margin:0 4em"><a href="/wiki/Banach_space" title="Banach space">Banach space</a> topics</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of Banach spaces</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Asplund_space" title="Asplund space">Asplund</a></li> <li><a href="/wiki/Banach_space" title="Banach space">Banach</a> <ul><li><a href="/wiki/List_of_Banach_spaces" title="List of Banach spaces">list</a></li></ul></li> <li><a href="/wiki/Banach_lattice" title="Banach lattice">Banach lattice</a></li> <li><a href="/wiki/Grothendieck_space" title="Grothendieck space">Grothendieck </a></li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert</a> <ul><li><a href="/wiki/Inner_product_space" title="Inner product space">Inner product space</a></li> <li><a href="/wiki/Polarization_identity" title="Polarization identity">Polarization identity</a></li></ul></li> <li>(<a href="/wiki/Polynomially_reflexive_space" title="Polynomially reflexive space">Polynomially</a>) <a href="/wiki/Reflexive_space" title="Reflexive space">Reflexive</a></li> <li><a href="/wiki/Riesz_space" title="Riesz space">Riesz</a></li> <li><a href="/wiki/L-semi-inner_product" title="L-semi-inner product">L-semi-inner product</a></li> <li>(<a href="/wiki/B-convex_space" title="B-convex space">B</a></li> <li><a href="/wiki/Strictly_convex_space" title="Strictly convex space">Strictly</a></li> <li><a href="/wiki/Uniformly_convex_space" title="Uniformly convex space">Uniformly</a>) convex</li> <li><a href="/wiki/Uniformly_smooth_space" title="Uniformly smooth space">Uniformly smooth</a></li> <li>(<a href="/wiki/Injective_tensor_product" title="Injective tensor product">Injective</a></li> <li><a href="/wiki/Projective_tensor_product" title="Projective tensor product">Projective</a>) <a href="/wiki/Topological_tensor_product" title="Topological tensor product">Tensor product</a> (<a href="/wiki/Tensor_product_of_Hilbert_spaces" title="Tensor product of Hilbert spaces">of Hilbert spaces</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Banach spaces are:</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Barrelled_space" title="Barrelled space">Barrelled</a></li> <li><a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">Complete</a></li> <li><a href="/wiki/F-space" title="F-space">F-space</a></li> <li><a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet</a> <ul><li><a href="/wiki/Differentiation_in_Fr%C3%A9chet_spaces#Tame_Fréchet_spaces" title="Differentiation in Fréchet spaces">tame</a></li></ul></li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex</a> <ul><li><a href="/wiki/Locally_convex_topological_vector_space#Definition_via_seminorms" title="Locally convex topological vector space">Seminorms</a>/<a href="/wiki/Minkowski_functional" title="Minkowski functional">Minkowski functionals</a></li></ul></li> <li><a href="/wiki/Mackey_space" title="Mackey space">Mackey</a></li> <li><a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">Metrizable</a></li> <li><a href="/wiki/Normed_space" class="mw-redirect" title="Normed space">Normed</a> <ul><li><a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a></li></ul></li> <li><a href="/wiki/Quasinorm" title="Quasinorm">Quasinormed</a></li> <li><a href="/wiki/Stereotype_space" class="mw-redirect" title="Stereotype space">Stereotype</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Function space Topologies</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach%E2%80%93Mazur_compactum" title="Banach–Mazur compactum">Banach–Mazur compactum</a></li> <li><a href="/wiki/Dual_topology" title="Dual topology">Dual</a></li> <li><a href="/wiki/Dual_space" title="Dual space">Dual space</a> <ul><li><a href="/wiki/Dual_norm" title="Dual norm">Dual norm</a></li></ul></li> <li><a href="/wiki/Operator_topologies" title="Operator topologies">Operator</a></li> <li><a href="/wiki/Ultraweak_topology" title="Ultraweak topology">Ultraweak</a></li> <li><a href="/wiki/Weak_topology" title="Weak topology">Weak</a> <ul><li><a href="/wiki/Weak_topology_(polar_topology)" class="mw-redirect" title="Weak topology (polar topology)">polar</a></li> <li><a href="/wiki/Weak_operator_topology" title="Weak operator topology">operator</a></li></ul></li> <li><a href="/wiki/Strong_topology" title="Strong topology">Strong</a> <ul><li><a href="/wiki/Strong_topology_(polar_topology)" class="mw-redirect" title="Strong topology (polar topology)">polar</a></li> <li><a href="/wiki/Strong_operator_topology" title="Strong operator topology">operator</a></li></ul></li> <li><a href="/wiki/Ultrastrong_topology" title="Ultrastrong topology">Ultrastrong</a></li> <li><a href="/wiki/Topology_of_uniform_convergence" class="mw-redirect" title="Topology of uniform convergence">Uniform convergence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">Linear operators</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hermitian_adjoint" title="Hermitian adjoint">Adjoint</a></li> <li><a href="/wiki/Bilinear_map" title="Bilinear map">Bilinear</a> <ul><li><a href="/wiki/Bilinear_form" title="Bilinear form">form</a></li> <li><a href="/wiki/Bilinear_map" title="Bilinear map">operator</a></li> <li><a href="/wiki/Sesquilinear_form" title="Sesquilinear form">sesquilinear</a></li></ul></li> <li>(<a href="/wiki/Unbounded_operator" title="Unbounded operator">Un</a>)<a href="/wiki/Bounded_operator" title="Bounded operator">Bounded</a></li> <li><a href="/wiki/Closed_linear_operator" title="Closed linear operator">Closed</a></li> <li><a href="/wiki/Compact_operator" title="Compact operator">Compact</a> <ul><li><a href="/wiki/Compact_operator_on_Hilbert_space" title="Compact operator on Hilbert space">on Hilbert spaces</a></li></ul></li> <li>(<a href="/wiki/Discontinuous_linear_map" title="Discontinuous linear map">Dis</a>)<a href="/wiki/Continuous_linear_operator" title="Continuous linear operator">Continuous</a></li> <li><a href="/wiki/Densely_defined" class="mw-redirect" title="Densely defined">Densely defined</a></li> <li>Fredholm <ul><li><a href="/wiki/Fredholm_kernel" title="Fredholm kernel">kernel</a></li> <li><a href="/wiki/Fredholm_operator" title="Fredholm operator">operator</a></li></ul></li> <li><a href="/wiki/Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt</a></li> <li><a href="/wiki/Linear_form" title="Linear form">Functionals</a> <ul><li><a href="/wiki/Positive_linear_functional" title="Positive linear functional">positive</a></li></ul></li> <li><a href="/wiki/Pseudo-monotone_operator" title="Pseudo-monotone operator">Pseudo-monotone</a></li> <li><a href="/wiki/Normal_operator" title="Normal operator">Normal</a></li> <li><a href="/wiki/Nuclear_operator" title="Nuclear operator">Nuclear</a></li> <li><a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">Self-adjoint</a></li> <li><a href="/wiki/Strictly_singular_operator" title="Strictly singular operator">Strictly singular</a></li> <li><a href="/wiki/Trace_class" title="Trace class">Trace class</a></li> <li><a href="/wiki/Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose</a></li> <li><a href="/wiki/Unitary_operator" title="Unitary operator">Unitary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Operator_theory" title="Operator theory">Operator theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_algebra" title="Banach algebra">Banach algebras</a></li> <li><a href="/wiki/C*-algebra" title="C*-algebra">C*-algebras</a></li> <li><a href="/wiki/Operator_space" title="Operator space">Operator space</a></li> <li><a href="/wiki/Spectrum_(functional_analysis)" title="Spectrum (functional analysis)">Spectrum</a> <ul><li><a href="/wiki/Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">C*-algebra</a></li> <li><a href="/wiki/Spectral_radius" title="Spectral radius">radius</a></li></ul></li> <li><a href="/wiki/Spectral_theory" title="Spectral theory">Spectral theory</a> <ul><li><a href="/wiki/Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">of ODEs</a></li> <li><a href="/wiki/Spectral_theorem" title="Spectral theorem">Spectral theorem</a></li></ul></li> <li><a href="/wiki/Polar_decomposition" title="Polar decomposition">Polar decomposition</a></li> <li><a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">Singular value decomposition</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Theorems_in_functional_analysis" title="Category:Theorems in functional analysis">Theorems</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Anderson%E2%80%93Kadec_theorem" title="Anderson–Kadec theorem">Anderson–Kadec</a></li> <li><a href="/wiki/Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu</a></li> <li><a href="/wiki/Banach%E2%80%93Mazur_theorem" title="Banach–Mazur theorem">Banach–Mazur</a></li> <li><a href="/wiki/Banach%E2%80%93Saks_theorem" class="mw-redirect" title="Banach–Saks theorem">Banach–Saks</a></li> <li><a href="/wiki/Open_mapping_theorem_(functional_analysis)" title="Open mapping theorem (functional analysis)">Banach–Schauder (open mapping)</a></li> <li><a href="/wiki/Uniform_boundedness_principle" title="Uniform boundedness principle">Banach–Steinhaus (Uniform boundedness)</a></li> <li><a href="/wiki/Bessel%27s_inequality" title="Bessel's inequality">Bessel's inequality</a></li> <li><a href="/wiki/Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz inequality</a></li> <li><a href="/wiki/Closed_graph_theorem" title="Closed graph theorem">Closed graph</a></li> <li><a href="/wiki/Closed_range_theorem" title="Closed range theorem">Closed range</a></li> <li><a href="/wiki/Eberlein%E2%80%93%C5%A0mulian_theorem" title="Eberlein–Šmulian theorem">Eberlein–Šmulian</a></li> <li><a href="/wiki/Freudenthal_spectral_theorem" title="Freudenthal spectral theorem">Freudenthal spectral</a></li> <li><a href="/wiki/Gelfand%E2%80%93Mazur_theorem" title="Gelfand–Mazur theorem">Gelfand–Mazur</a></li> <li><a href="/wiki/Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark</a></li> <li><a href="/wiki/Goldstine_theorem" title="Goldstine theorem">Goldstine</a></li> <li><a href="/wiki/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a> <ul><li><a href="/wiki/Hyperplane_separation_theorem" title="Hyperplane separation theorem">hyperplane separation</a></li></ul></li> <li><a href="/wiki/Kakutani_fixed-point_theorem#Infinite-dimensional_generalizations" title="Kakutani fixed-point theorem">Kakutani fixed-point</a></li> <li><a href="/wiki/Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman</a></li> <li><a href="/wiki/Invariant_subspace_problem#Known_special_cases" title="Invariant subspace problem">Lomonosov's invariant subspace</a></li> <li><a href="/wiki/Mackey%E2%80%93Arens_theorem" title="Mackey–Arens theorem">Mackey–Arens</a></li> <li><a href="/wiki/Mazur%27s_lemma" title="Mazur's lemma">Mazur's lemma</a></li> <li><a href="/wiki/M._Riesz_extension_theorem" title="M. Riesz extension theorem">M. Riesz extension</a></li> <li><a href="/wiki/Parseval%27s_identity" title="Parseval's identity">Parseval's identity</a></li> <li><a href="/wiki/Riesz%27s_lemma" title="Riesz's lemma">Riesz's lemma</a></li> <li><a href="/wiki/Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation</a></li> <li><a href="/wiki/Ursescu_theorem#Robinson–Ursescu_theorem" title="Ursescu theorem">Robinson-Ursescu</a></li> <li><a href="/wiki/Schauder_fixed-point_theorem" title="Schauder fixed-point theorem">Schauder fixed-point</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Analysis</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_Wiener_space" title="Abstract Wiener space">Abstract Wiener space</a></li> <li><a class="mw-selflink selflink">Banach manifold</a> <ul><li><a href="/wiki/Banach_bundle" title="Banach bundle">bundle</a></li></ul></li> <li><a href="/wiki/Bochner_space" title="Bochner space">Bochner space</a></li> <li><a href="/wiki/Convex_series" title="Convex series">Convex series</a></li> <li><a href="/wiki/Differentiation_in_Fr%C3%A9chet_spaces" title="Differentiation in Fréchet spaces">Differentiation in Fréchet spaces</a></li> <li><a href="/wiki/Derivative" title="Derivative">Derivatives</a> <ul><li><a href="/wiki/Fr%C3%A9chet_derivative" title="Fréchet derivative">Fréchet</a></li> <li><a href="/wiki/Gateaux_derivative" title="Gateaux derivative">Gateaux</a></li> <li><a href="/wiki/Functional_derivative" title="Functional derivative">functional</a></li> <li><a href="/wiki/Infinite-dimensional_holomorphy" title="Infinite-dimensional holomorphy">holomorphic</a></li> <li><a href="/wiki/Quasi-derivative" title="Quasi-derivative">quasi</a></li></ul></li> <li><a href="/wiki/Integral" title="Integral">Integrals</a> <ul><li><a href="/wiki/Bochner_integral" title="Bochner integral">Bochner</a></li> <li><a href="/wiki/Dunford_integral" class="mw-redirect" title="Dunford integral">Dunford</a></li> <li><a href="/wiki/Pettis_integral" title="Pettis integral">Gelfand–Pettis</a></li> <li><a href="/wiki/Regulated_integral" title="Regulated integral">regulated</a></li> <li><a href="/wiki/Paley%E2%80%93Wiener_integral" title="Paley–Wiener integral">Paley–Wiener</a></li> <li><a href="/wiki/Pettis_integral" title="Pettis integral">weak</a></li></ul></li> <li><a href="/wiki/Functional_calculus" title="Functional calculus">Functional calculus</a> <ul><li><a href="/wiki/Borel_functional_calculus" title="Borel functional calculus">Borel</a></li> <li><a href="/wiki/Continuous_functional_calculus" title="Continuous functional calculus">continuous</a></li> <li><a href="/wiki/Holomorphic_functional_calculus" title="Holomorphic functional calculus">holomorphic</a></li></ul></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measures</a> <ul><li><a href="/wiki/Infinite-dimensional_Lebesgue_measure" title="Infinite-dimensional Lebesgue measure">Lebesgue</a></li> <li><a href="/wiki/Projection-valued_measure" title="Projection-valued measure">Projection-valued</a></li> <li><a href="/wiki/Vector_measure" title="Vector measure">Vector</a></li></ul></li> <li><a href="/wiki/Weakly_measurable_function" title="Weakly measurable function">Weakly</a> / <a href="/wiki/Strongly_measurable_functions" class="mw-redirect" title="Strongly measurable functions">Strongly</a> measurable function</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of sets</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Absolutely_convex_set" title="Absolutely convex set">Absolutely convex</a></li> <li><a href="/wiki/Absorbing_set" title="Absorbing set">Absorbing</a></li> <li><a href="/wiki/Affine_space" title="Affine space">Affine</a></li> <li><a href="/wiki/Balanced_set" title="Balanced set">Balanced/Circled</a></li> <li><a href="/wiki/Bounded_set_(topological_vector_space)" title="Bounded set (topological vector space)">Bounded</a></li> <li><a href="/wiki/Convex_set" title="Convex set">Convex</a></li> <li><a href="/wiki/Convex_cone" title="Convex cone">Convex cone <span style="font-size:85%;">(subset)</span></a></li> <li><a href="/wiki/Convex_series#Types_of_subsets" title="Convex series">Convex series related</a> ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (H<i>x</i>), and (Hw<i>x</i>))</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/Radial_set" title="Radial set">Radial</a></li> <li><a href="/wiki/Star_domain" title="Star domain">Radially convex/Star-shaped</a></li> <li><a href="/wiki/Symmetric_set" title="Symmetric set">Symmetric</a></li> <li><a href="/wiki/Zonotope" class="mw-redirect" title="Zonotope">Zonotope</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Subsets / set operations</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_hull" title="Affine hull">Affine hull</a></li> <li>(<a href="/wiki/Algebraic_interior#Relative_algebraic_interior" title="Algebraic interior">Relative</a>) <a href="/wiki/Algebraic_interior" title="Algebraic interior">Algebraic interior (core)</a></li> <li><a href="/wiki/Bounding_point" title="Bounding point">Bounding points</a></li> <li><a href="/wiki/Convex_hull" title="Convex hull">Convex hull</a></li> <li><a href="/wiki/Extreme_point" title="Extreme point">Extreme point</a></li> <li><a href="/wiki/Interior_(topology)" title="Interior (topology)">Interior</a></li> <li><a href="/wiki/Linear_span" title="Linear span">Linear span</a></li> <li><a href="/wiki/Minkowski_addition" title="Minkowski addition">Minkowski addition</a></li> <li><a href="/wiki/Polar_set" title="Polar set">Polar</a></li> <li>(<a href="/wiki/Algebraic_interior#Quasi_relative_interior" title="Algebraic interior">Quasi</a>) <a href="/wiki/Algebraic_interior#Relative_interior" title="Algebraic interior">Relative interior</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Template:ListOfBanachSpaces" class="mw-redirect" title="Template:ListOfBanachSpaces">Examples</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Absolute_continuity" title="Absolute continuity">Absolute continuity <i>AC</i></a></li> <li><a href="/wiki/Ba_space" title="Ba space"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ba(\Sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mi>a</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ba(\Sigma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58fe61351e3531b14043fa2d09e98c2437bd1a6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.715ex; height:2.843ex;" alt="{\displaystyle ba(\Sigma )}" /></span></a></li> <li><a href="/wiki/C_space" title="C space">c space</a></li> <li><a href="/wiki/BK-space" title="BK-space">Banach coordinate <i>BK</i></a></li> <li><a href="/wiki/Besov_space" title="Besov space">Besov <span class="mwe-math-element"><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_{p,q}^{s}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{p,q}^{s}(\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9919cf78ad095c237169772d2b27a37bfbef1b75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.524ex; height:3.009ex;" alt="{\displaystyle B_{p,q}^{s}(\mathbb {R} )}" /></span></a></li> <li><a href="/wiki/Birnbaum%E2%80%93Orlicz_space" class="mw-redirect" title="Birnbaum–Orlicz space">Birnbaum–Orlicz</a></li> <li><a href="/wiki/Bounded_variation" title="Bounded variation">Bounded variation <i>BV</i></a></li> <li><a href="/wiki/Bs_space" title="Bs space">Bs space</a></li> <li><a href="/wiki/Continuous_functions_on_a_compact_Hausdorff_space" title="Continuous functions on a compact Hausdorff space">Continuous <i>C(K)</i> with <i>K</i> compact Hausdorff</a></li> <li><a href="/wiki/Hardy_space" title="Hardy space">Hardy H<sup><i>p</i></sup></a></li> <li><a href="/wiki/Hilbert_space#Definition" title="Hilbert space">Hilbert <i>H</i></a></li> <li><a href="/wiki/Morrey%E2%80%93Campanato_space" title="Morrey–Campanato space">Morrey–Campanato <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\lambda ,p}(\Omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo>,</mo> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\lambda ,p}(\Omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b8af58fa038369c3ec6386c6656aab82825e372" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.545ex; height:3.176ex;" alt="{\displaystyle L^{\lambda ,p}(\Omega )}" /></span></a></li> <li><a href="/wiki/Sequence_space#ℓp_spaces" title="Sequence space"><i>ℓ<sup>p</sup></i></a> <ul><li><a href="/wiki/L-infinity#Sequence_space" title="L-infinity"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8348195cf09473662c6f59e6717722a6fc01d0f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.845ex; height:2.343ex;" alt="{\displaystyle \ell ^{\infty }}" /></span></a></li></ul></li> <li><a href="/wiki/Lp_space" title="Lp space"><i>L<sup>p</sup></i></a> <ul><li><a href="/wiki/L-infinity#Function_space" title="L-infinity"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ab400cc4dfd865180cd84c72dc894ca457671f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.458ex; height:2.343ex;" alt="{\displaystyle L^{\infty }}" /></span></a></li> <li><a href="/wiki/Lp_space#Weighted_Lp_spaces" title="Lp space">weighted</a></li></ul></li> <li><a href="/wiki/Schwartz_space" title="Schwartz space">Schwartz <span class="mwe-math-element"><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\left(\mathbb {R} ^{n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\left(\mathbb {R} ^{n}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0465acd58a0f31e32b095aed742d9ccc6331369c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.592ex; height:2.843ex;" alt="{\displaystyle S\left(\mathbb {R} ^{n}\right)}" /></span></a></li> <li><a href="/wiki/Segal%E2%80%93Bargmann_space" title="Segal–Bargmann space">Segal–Bargmann <i>F</i></a></li> <li><a href="/wiki/Sequence_space" title="Sequence space">Sequence space</a></li> <li><a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev W<sup><i>k,p</i></sup></a> <ul><li><a href="/wiki/Sobolev_inequality" title="Sobolev inequality">Sobolev inequality</a></li></ul></li> <li><a href="/wiki/Triebel%E2%80%93Lizorkin_space" title="Triebel–Lizorkin space">Triebel–Lizorkin</a></li> <li><a href="/wiki/Wiener_amalgam_space" title="Wiener amalgam space">Wiener amalgam <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(X,L^{p})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W(X,L^{p})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b37b1dc9714960c525cb561a4828f41feb5844ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.9ex; height:2.843ex;" alt="{\displaystyle W(X,L^{p})}" /></span></a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differential_operator" title="Differential operator">Differential operator</a></li> <li><a href="/wiki/Finite_element_method" title="Finite element method">Finite element method</a></li> <li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Mathematical formulation of quantum mechanics</a></li> <li><a href="/wiki/Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Ordinary Differential Equations (ODEs)</a></li> <li><a href="/wiki/Validated_numerics" title="Validated numerics">Validated numerics</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /></div><div role="navigation" class="navbox" aria-labelledby="Functional_analysis_(topics_–_glossary)364" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231" /><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Functional_analysis" title="Template:Functional analysis"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Functional_analysis" title="Template talk:Functional analysis"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Functional_analysis" title="Special:EditPage/Template:Functional analysis"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Functional_analysis_(topics_–_glossary)364" style="font-size:114%;margin:0 4em"><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a> (<a href="/wiki/List_of_functional_analysis_topics" title="List of functional analysis topics">topics</a> – <a href="/wiki/Glossary_of_functional_analysis" title="Glossary of functional analysis">glossary</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Spaces</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_space" title="Banach space">Banach</a></li> <li><a href="/wiki/Besov_space" title="Besov space">Besov</a></li> <li><a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet</a></li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert</a></li> <li><a href="/wiki/H%C3%B6lder_space" class="mw-redirect" title="Hölder space">Hölder</a></li> <li><a href="/wiki/Nuclear_space" title="Nuclear space">Nuclear</a></li> <li><a href="/wiki/Orlicz_space" title="Orlicz space">Orlicz</a></li> <li><a href="/wiki/Schwartz_space" title="Schwartz space">Schwartz</a></li> <li><a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev</a></li> <li><a href="/wiki/Topological_vector_space" title="Topological vector space">Topological vector</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Barrelled_space" title="Barrelled space">Barrelled</a></li> <li><a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">Complete</a></li> <li><a href="/wiki/Dual_space" title="Dual space">Dual</a> (<a href="/wiki/Dual_space#Algebraic_dual_space" title="Dual space">Algebraic</a> / <a href="/wiki/Dual_space#Continuous_dual_space" title="Dual space">Topological</a>)</li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex</a></li> <li><a href="/wiki/Reflexive_space" title="Reflexive space">Reflexive</a></li> <li><a href="/wiki/Separable_space" title="Separable space">Separable</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Theorems_in_functional_analysis" title="Category:Theorems in functional analysis">Theorems</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a></li> <li><a href="/wiki/Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation</a></li> <li><a href="/wiki/Closed_graph_theorem_(functional_analysis)" title="Closed graph theorem (functional analysis)">Closed graph</a></li> <li><a href="/wiki/Uniform_boundedness_principle" title="Uniform boundedness principle">Uniform boundedness principle</a></li> <li><a href="/wiki/Kakutani_fixed-point_theorem#Infinite-dimensional_generalizations" title="Kakutani fixed-point theorem">Kakutani fixed-point</a></li> <li><a href="/wiki/Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman</a></li> <li><a href="/wiki/Min-max_theorem" title="Min-max theorem">Min–max</a></li> <li><a href="/wiki/Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark</a></li> <li><a href="/wiki/Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Operators</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adjoint_operator" class="mw-redirect" title="Adjoint operator">Adjoint</a></li> <li><a href="/wiki/Bounded_operator" title="Bounded operator">Bounded</a></li> <li><a href="/wiki/Compact_operator" title="Compact operator">Compact</a></li> <li><a href="/wiki/Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt</a></li> <li><a href="/wiki/Normal_operator" title="Normal operator">Normal</a></li> <li><a href="/wiki/Nuclear_operator" title="Nuclear operator">Nuclear</a></li> <li><a href="/wiki/Trace_class" title="Trace class">Trace class</a></li> <li><a href="/wiki/Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose</a></li> <li><a href="/wiki/Unbounded_operator" title="Unbounded operator">Unbounded</a></li> <li><a href="/wiki/Unitary_operator" title="Unitary operator">Unitary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebras</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_algebra" title="Banach algebra">Banach algebra</a></li> <li><a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a></li> <li><a href="/wiki/Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">Spectrum of a C*-algebra</a></li> <li><a href="/wiki/Operator_algebra" title="Operator algebra">Operator algebra</a></li> <li><a href="/wiki/Group_algebra_of_a_locally_compact_group" title="Group algebra of a locally compact group">Group algebra of a locally compact group</a></li> <li><a href="/wiki/Von_Neumann_algebra" title="Von Neumann algebra">Von Neumann algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Open problems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Invariant_subspace_problem" title="Invariant subspace problem">Invariant subspace problem</a></li> <li><a href="/wiki/Mahler%27s_conjecture" class="mw-redirect" title="Mahler's conjecture">Mahler's conjecture</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hardy_space" title="Hardy space">Hardy space</a></li> <li><a href="/wiki/Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Spectral theory of ordinary differential equations</a></li> <li><a href="/wiki/Heat_kernel" title="Heat kernel">Heat kernel</a></li> <li><a href="/wiki/Index_theorem" class="mw-redirect" title="Index theorem">Index theorem</a></li> <li><a href="/wiki/Calculus_of_variations" title="Calculus of variations">Calculus of variations</a></li> <li><a href="/wiki/Functional_calculus" title="Functional calculus">Functional calculus</a></li> <li><a href="/wiki/Integral_linear_operator" title="Integral linear operator">Integral linear operator</a></li> <li><a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones polynomial</a></li> <li><a href="/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">Topological quantum field theory</a></li> <li><a href="/wiki/Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a></li> <li><a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></li> <li><a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">Distribution</a> (or <a href="/wiki/Generalized_function" title="Generalized function">Generalized functions</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Advanced topics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Approximation_property" title="Approximation property">Approximation property</a></li> <li><a href="/wiki/Balanced_set" title="Balanced set">Balanced set</a></li> <li><a href="/wiki/Choquet_theory" title="Choquet theory">Choquet theory</a></li> <li><a href="/wiki/Weak_topology" title="Weak topology">Weak topology</a></li> <li><a href="/wiki/Banach%E2%80%93Mazur_distance" class="mw-redirect" title="Banach–Mazur distance">Banach–Mazur distance</a></li> <li><a href="/wiki/Tomita%E2%80%93Takesaki_theory" title="Tomita–Takesaki theory">Tomita–Takesaki theory</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Functional_analysis" title="Category:Functional analysis">Category</a></li></ul> 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