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Strong dual space
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<h2 id="strong-dual-topology">Strong dual topology</h2> <p>Throughout, all vector spaces will be assumed to be over the field F {\displaystyle \mathbb {F} } of either the <a href="/facts/Real_number/R02gw5Pb">real numbers</a> R {\displaystyle \mathbb {R} } or <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> C . {\displaystyle \mathbb {C} .} </p> <h3>Definition from a dual system</h3> <p class="note">Main article: <a href="/facts/Dual_system/2Y8KyofV">Dual system</a></p> <p>Let ( X , Y , ⟨ ⋅ , ⋅ ⟩ ) {\displaystyle (X,Y,\langle \cdot ,\cdot \rangle )} be a <a href="/facts/Dual_pair/2Y8KyofV">dual pair</a> of vector spaces over the field F {\displaystyle \mathbb {F} } of <a href="/facts/Real_number/R02gw5Pb">real numbers</a> R {\displaystyle \mathbb {R} } or <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> C . {\displaystyle \mathbb {C} .} For any B ⊆ X {\displaystyle B\subseteq X} and any y ∈ Y , {\displaystyle y\in Y,} define | y | B = sup x ∈ B | ⟨ x , y ⟩ | . {\displaystyle |y|_{B}=\sup _{x\in B}|\langle x,y\rangle |.} </p><p>Neither X {\displaystyle X} nor Y {\displaystyle Y} has a topology so say a subset B ⊆ X {\displaystyle B\subseteq X} is said to be bounded by a subset C ⊆ Y {\displaystyle C\subseteq Y} if | y | B < ∞ {\displaystyle |y|_{B}<\infty } for all y ∈ C . {\displaystyle y\in C.} So a subset B ⊆ X {\displaystyle B\subseteq X} is called bounded if and only if sup x ∈ B | ⟨ x , y ⟩ | < ∞ for all y ∈ Y . {\displaystyle \sup _{x\in B}|\langle x,y\rangle |<\infty \quad {\text{ for all }}y\in Y.} This is equivalent to the usual notion of <a href="/facts/Bounded_set_(topological_vector_space)/bTsWeU9g">bounded subsets</a> when X {\displaystyle X} is given the weak topology induced by Y , {\displaystyle Y,} which is a Hausdorff <a href="/facts/Locally_convex/AGrPrayC">locally convex</a> topology. </p><p>Let B {\displaystyle {\mathcal {B}}} denote the <a href="/facts/Family_of_sets/WbYof7lP">family</a> of all subsets B ⊆ X {\displaystyle B\subseteq X} bounded by elements of Y {\displaystyle Y} ; that is, B {\displaystyle {\mathcal {B}}} is the set of all subsets B ⊆ X {\displaystyle B\subseteq X} such that for every y ∈ Y , {\displaystyle y\in Y,} | y | B = sup x ∈ B | ⟨ x , y ⟩ | < ∞ . {\displaystyle |y|_{B}=\sup _{x\in B}|\langle x,y\rangle |<\infty .} Then the strong topology β ( Y , X , ⟨ ⋅ , ⋅ ⟩ ) {\displaystyle \beta (Y,X,\langle \cdot ,\cdot \rangle )} on Y , {\displaystyle Y,} also denoted by b ( Y , X , ⟨ ⋅ , ⋅ ⟩ ) {\displaystyle b(Y,X,\langle \cdot ,\cdot \rangle )} or simply β ( Y , X ) {\displaystyle \beta (Y,X)} or b ( Y , X ) {\displaystyle b(Y,X)} if the pairing ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is understood, is defined as the <a href="/facts/Locally_convex_topological_vector_space/AGrPrayC">locally convex</a> topology on Y {\displaystyle Y} generated by the seminorms of the form | y | B = sup x ∈ B | ⟨ x , y ⟩ | , y ∈ Y , B ∈ B . {\displaystyle |y|_{B}=\sup _{x\in B}|\langle x,y\rangle |,\qquad y\in Y,\qquad B\in {\mathcal {B}}.} </p><p>The definition of the strong dual topology now proceeds as in the case of a TVS. Note that if X {\displaystyle X} is a TVS whose continuous dual space <a href="/facts/Separating_set/I8VjYAgQ">separates points</a> on X , {\displaystyle X,} then X {\displaystyle X} is part of a canonical dual system ( X , X ′ , ⟨ ⋅ , ⋅ ⟩ ) {\displaystyle \left(X,X^{\prime },\langle \cdot ,\cdot \rangle \right)} where ⟨ x , x ′ ⟩ := x ′ ( x ) . {\displaystyle \left\langle x,x^{\prime }\right\rangle :=x^{\prime }(x).} In the special case when X {\displaystyle X} is a <a href="/facts/Locally_convex_space/AGrPrayC">locally convex space</a>, the strong topology on the (continuous) <a href="/facts/Continuous_dual_space/4Uw4Knz1">dual space</a> X ′ {\displaystyle X^{\prime }} (that is, on the space of all continuous linear functionals f : X → F {\displaystyle f:X\to \mathbb {F} } ) is defined as the strong topology β ( X ′ , X ) , {\displaystyle \beta \left(X^{\prime },X\right),} and it coincides with the topology of uniform convergence on <a href="/facts/Bounded_set/yCldjtoy">bounded sets</a> in X , {\displaystyle X,} i.e. with the topology on X ′ {\displaystyle X^{\prime }} generated by the seminorms of the form | f | B = sup x ∈ B | f ( x ) | , where f ∈ X ′ , {\displaystyle |f|_{B}=\sup _{x\in B}|f(x)|,\qquad {\text{ where }}f\in X^{\prime },} where B {\displaystyle B} runs over the family of all <a href="/facts/Bounded_set_(topological_vector_space)/bTsWeU9g">bounded sets</a> in X . {\displaystyle X.} The space X ′ {\displaystyle X^{\prime }} with this topology is called strong dual space of the space X {\displaystyle X} and is denoted by X β ′ . {\displaystyle X_{\beta }^{\prime }.} </p> <h3>Definition on a TVS</h3> <p>Suppose that X {\displaystyle X} is a <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> (TVS) over the field F . {\displaystyle \mathbb {F} .} Let B {\displaystyle {\mathcal {B}}} be any fundamental system of <a href="/facts/Bounded_set_(topological_vector_space)/bTsWeU9g">bounded sets</a> of X {\displaystyle X} ; that is, B {\displaystyle {\mathcal {B}}} is a <a href="/facts/Family_of_sets/WbYof7lP">family</a> of bounded subsets of X {\displaystyle X} such that every bounded subset of X {\displaystyle X} is a subset of some B ∈ B {\displaystyle B\in {\mathcal {B}}} ; the set of all bounded subsets of X {\displaystyle X} forms a fundamental system of bounded sets of X . {\displaystyle X.} A basis of closed neighborhoods of the origin in X ′ {\displaystyle X^{\prime }} is given by the <a href="/facts/Polar_set/JY45pfbt">polars</a>: B ∘ := { x ′ ∈ X ′ : sup x ∈ B | x ′ ( x ) | ≤ 1 } {\displaystyle B^{\circ }:=\left\{x^{\prime }\in X^{\prime }:\sup _{x\in B}\left|x^{\prime }(x)\right|\leq 1\right\}} as B {\displaystyle B} ranges over B {\displaystyle {\mathcal {B}}} ). This is a locally convex topology that is given by the set of <a href="/facts/Seminorm/vMuescKp">seminorms</a> on X ′ {\displaystyle X^{\prime }} : | x ′ | B := sup x ∈ B | x ′ ( x ) | {\displaystyle \left|x^{\prime }\right|_{B}:=\sup _{x\in B}\left|x^{\prime }(x)\right|} as B {\displaystyle B} ranges over B . {\displaystyle {\mathcal {B}}.} </p><p>If X {\displaystyle X} is <a href="/facts/Normable_space/rmDljfyE">normable</a> then so is X b ′ {\displaystyle X_{b}^{\prime }} and X b ′ {\displaystyle X_{b}^{\prime }} will in fact be a <a href="/facts/Banach_space/sDEIk7EC">Banach space</a>. If X {\displaystyle X} is a normed space with norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} then X ′ {\displaystyle X^{\prime }} has a canonical norm (the <a href="/facts/Operator_norm/a5l0TkFW">operator norm</a>) given by ‖ x ′ ‖ := sup ‖ x ‖ ≤ 1 | x ′ ( x ) | {\displaystyle \left\|x^{\prime }\right\|:=\sup _{\|x\|\leq 1}\left|x^{\prime }(x)\right|} ; the topology that this norm induces on X ′ {\displaystyle X^{\prime }} is identical to the strong dual topology. </p> <h2 id="bidual">Bidual</h2> <p class="note">See also: <a href="/facts/Banach_space/sDEIk7EC">Banach space § Bidual</a>, <a href="/facts/Reflexive_space/g7Cjbw0S">Reflexive space</a>, <a href="/facts/Semi-reflexive_space/NRcUlayR">Semi-reflexive space</a>, and <a href="/facts/Double_dual/4Uw4Knz1">Double dual</a></p> <p>The bidual or second dual of a TVS X , {\displaystyle X,} often denoted by X ′ ′ , {\displaystyle X^{\prime \prime },} is the strong dual of the strong dual of X {\displaystyle X} : X ′ ′ := ( X b ′ ) ′ {\displaystyle X^{\prime \prime }\,:=\,\left(X_{b}^{\prime }\right)^{\prime }} where X b ′ {\displaystyle X_{b}^{\prime }} denotes X ′ {\displaystyle X^{\prime }} endowed with the strong dual topology b ( X ′ , X ) . {\displaystyle b\left(X^{\prime },X\right).} Unless indicated otherwise, the vector space X ′ ′ {\displaystyle X^{\prime \prime }} is usually assumed to be endowed with the strong dual topology induced on it by X b ′ , {\displaystyle X_{b}^{\prime },} in which case it is called the strong bidual of X {\displaystyle X} ; that is, X ′ ′ := ( X b ′ ) b ′ {\displaystyle X^{\prime \prime }\,:=\,\left(X_{b}^{\prime }\right)_{b}^{\prime }} where the vector space X ′ ′ {\displaystyle X^{\prime \prime }} is endowed with the strong dual topology b ( X ′ ′ , X b ′ ) . {\displaystyle b\left(X^{\prime \prime },X_{b}^{\prime }\right).} </p> <h2 id="properties">Properties</h2> <p>Let X {\displaystyle X} be a <a href="/facts/Locally_convex_topological_vector_space/AGrPrayC">locally convex</a> TVS. </p> <ul><li>A convex <a href="/facts/Balanced_set/Y2ar7E6g">balanced</a> weakly compact subset of X ′ {\displaystyle X^{\prime }} is bounded in X b ′ . {\displaystyle X_{b}^{\prime }.} <a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a></li> <li>Every weakly bounded subset of X ′ {\displaystyle X^{\prime }} is strongly bounded.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></li> <li>If X {\displaystyle X} is a <a href="/facts/Barreled_space/r2H58TN3">barreled space</a> then X {\displaystyle X} 's topology is identical to the strong dual topology b ( X , X ′ ) {\displaystyle b\left(X,X^{\prime }\right)} and to the <a href="/facts/Mackey_topology/KI5zwcXv">Mackey topology</a> on X . {\displaystyle X.} </li> <li>If X {\displaystyle X} is a metrizable locally convex space, then the strong dual of X {\displaystyle X} is a <a href="/facts/Bornological_space/kpl5B2Ez">bornological space</a> if and only if it is an <a href="/facts/Infrabarreled_space/SaoLAH83">infrabarreled space</a>, if and only if it is a <a href="/facts/Barreled_space/r2H58TN3">barreled space</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></li> <li>If X {\displaystyle X} is Hausdorff locally convex TVS then ( X , b ( X , X ′ ) ) {\displaystyle \left(X,b\left(X,X^{\prime }\right)\right)} is <a href="/facts/Metrizable_topological_vector_space/6FUHfx6W">metrizable</a> if and only if there exists a countable set B {\displaystyle {\mathcal {B}}} of bounded subsets of X {\displaystyle X} such that every bounded subset of X {\displaystyle X} is contained in some element of B . {\displaystyle {\mathcal {B}}.} <a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></li> <li>If X {\displaystyle X} is locally convex, then this topology is finer than all other <a href="/facts/Topologies_on_spaces_of_linear_maps/TVmUH4ao"> G {\displaystyle {\mathcal {G}}} -topologies</a> on X ′ {\displaystyle X^{\prime }} when considering only G {\displaystyle {\mathcal {G}}} 's whose sets are subsets of X . {\displaystyle X.} </li> <li>If X {\displaystyle X} is a <a href="/facts/Bornological_space/kpl5B2Ez">bornological space</a> (e.g. <a href="/facts/Metrizable/BCLdo5Jf">metrizable</a> or <a href="/facts/LF-space/voQHEeGI">LF-space</a>) then X b ( X ′ , X ) ′ {\displaystyle X_{b(X^{\prime },X)}^{\prime }} is <a href="/facts/Complete_topological_vector_space/e6qtS76j">complete</a>.</li></ul> <p>If X {\displaystyle X} is a <a href="/facts/Barrelled_space/r2H58TN3">barrelled space</a>, then its topology coincides with the strong topology β ( X , X ′ ) {\displaystyle \beta \left(X,X^{\prime }\right)} on X {\displaystyle X} and with the <a href="/facts/Mackey_topology/KI5zwcXv">Mackey topology</a> on generated by the pairing ( X , X ′ ) . {\displaystyle \left(X,X^{\prime }\right).} </p> <h2 id="examples">Examples</h2> <p>If X {\displaystyle X} is a <a href="/facts/Normed_vector_space/rmDljfyE">normed vector space</a>, then its <a href="/facts/Continuous_dual/4Uw4Knz1">(continuous) dual space</a> X ′ {\displaystyle X^{\prime }} with the strong topology coincides with the <a href="/facts/Banach_space/sDEIk7EC">Banach dual space</a> X ′ {\displaystyle X^{\prime }} ; that is, with the space X ′ {\displaystyle X^{\prime }} with the topology induced by the <a href="/facts/Operator_norm/a5l0TkFW">operator norm</a>. Conversely ( X , X ′ ) . {\displaystyle \left(X,X^{\prime }\right).} -topology on X {\displaystyle X} is identical to the topology induced by the <a href="/facts/Norm_(mathematics)/xIbR4uE1">norm</a> on X . {\displaystyle X.} </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Dual_topology/DKPD7nYJ">Dual topology</a></li> <li><a href="/facts/Dual_system/2Y8KyofV">Dual system</a></li> <li><a href="/facts/List_of_topologies/mb4VoHSD">List of topologies</a> – List of concrete topologies and topological spaces</li> <li><a href="/facts/Polar_topology/WutbMK1V">Polar topology</a> – Dual space topology of uniform convergence on some sub-collection of bounded subsets</li> <li><a href="/facts/Reflexive_space/g7Cjbw0S">Reflexive space</a> – Locally convex topological vector space</li> <li><a href="/facts/Semi-reflexive_space/NRcUlayR">Semi-reflexive space</a></li> <li><a href="/facts/Strong_topology/vLDsdT18">Strong topology</a></li> <li><a href="/facts/Topologies_on_spaces_of_linear_maps/TVmUH4ao">Topologies on spaces of linear maps</a></li></ul> <h2 id="bibliography">Bibliography</h2> <ul><li>Narici, Lawrence; Beckenstein, Edward (2011). <i>Topological Vector Spaces</i>. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1584888666. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/144216834">144216834</a>.</li> <li><a href="/facts/Walter_Rudin/9mxRAECY">Rudin, Walter</a> (1991). <a href="https://archive.org/details/functionalanalys00rudi"><i>Functional Analysis</i></a>. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: <a href="/facts/McGraw-Hill_Science%2fEngineering%2fMath/srSyQSsY">McGraw-Hill Science/Engineering/Math</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-07-054236-5. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/21163277">21163277</a>.</li> <li><a href="/facts/Helmut_H._Schaefer/xorHTDBE">Schaefer, Helmut H.</a>; Wolff, Manfred P. (1999). <i>Topological Vector Spaces</i>. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">GTM</a>. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4612-7155-0. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/840278135">840278135</a>.</li> <li><a href="/facts/Fran%25C3%25A7ois_Tr%25C3%25A8ves/cpjQbAov">Trèves, François</a> (2006) [1967]. <i>Topological Vector Spaces, Distributions and Kernels</i>. Mineola, N.Y.: Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-45352-1. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/853623322">853623322</a>.</li> <li>Wong (1979). <i>Schwartz spaces, nuclear spaces, and tensor products</i>. Berlin New York: Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-09513-6. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/5126158">5126158</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Schaefer & Wolff 1999, p. 141. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Schaefer & Wolff 1999, p. 142. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Schaefer & Wolff 1999, p. 153. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Narici & Beckenstein 2011, pp. 225–273. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>