Editing Reflexive space (section)

=== Examples ===

<ol>
<li> Every finite-dimensional Hausdorff [[topological vector space]] is reflexive, because <math>J</math> is bijective by linear algebra, and because there is a unique Hausdorff vector space topology on a finite dimensional vector space.</li>
<li> A normed space <math>X</math> is reflexive as a normed space if and only if it is reflexive as a locally convex space. This follows from the fact that for a normed space <math>X</math> its dual normed space <math>X^{\prime}</math> coincides as a topological vector space with the strong dual space <math>X^{\prime}_b.</math> As a corollary, the evaluation map <math>J : X \to X^{\prime\prime}</math> coincides with the evaluation map <math>J : X \to \left(X^{\prime}_b\right)^{\prime}_b,</math> and the following conditions become equivalent:
<ol type="i">
<li><math>X</math> is a reflexive normed space (that is, <math>J : X \to X^{\prime\prime}</math> is an isomorphism of normed spaces),</li>
<li><math>X</math> is a reflexive locally convex space (that is, <math>J : X \to \left(X^{\prime}_b\right)^{\prime}_b</math> is an isomorphism of topological vector spaces<ref name=isomorphism />),</li>
<li><math>X</math> is a semi-reflexive locally convex space (that is, <math>J : X \to \left(X^{\prime}_b\right)^{\prime}_b</math> is surjective).</li>
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</li>
<li>A (somewhat artificial) example of a semi-reflexive space that is not reflexive is obtained as follows: let <math>Y</math> be an infinite dimensional reflexive Banach space, and let <math>X</math> be the topological vector space <math>\left(Y, \sigma\left(Y, Y^{\prime}\right)\right),</math> that is, the vector space <math>Y</math> equipped with the weak topology.  Then the continuous dual of <math>X</math> and <math>Y^{\prime}</math> are the same set of functionals, and bounded subsets of <math>X</math> (that is, weakly bounded subsets of <math>Y</math>) are norm-bounded, hence the Banach space <math>Y^{\prime}</math> is the strong dual of <math>X.</math> Since <math>Y</math> is reflexive, the continuous dual of <math>X^{\prime} = Y^{\prime}</math> is equal to the image <math>J(X)</math> of <math>X</math> under the canonical embedding <math>J,</math>  but the topology on <math>X</math> (the weak topology of <math>Y</math>) is not the strong topology <math>\beta\left(X, X^{\prime}\right),</math> that is equal to the norm topology of <math>Y.</math></li>
<li>[[Montel space]]s are reflexive locally convex topological vector spaces. In particular, the following functional spaces frequently used in functional analysis are reflexive locally convex spaces:<ref>{{harvnb|Edwards|1965|loc=8.4.7}}.</ref>
* the space <math>C^\infty(M)</math> of smooth functions on arbitrary (real) smooth manifold <math>M,</math> and its strong dual space <math>\left(C^\infty\right)^{\prime}(M)</math> of distributions with compact support on <math>M,</math>
* the space <math>\mathcal{D}(M)</math> of smooth functions with compact support on arbitrary (real) smooth manifold <math>M,</math> and its strong dual space <math>\mathcal{D}^{\prime}(M)</math> of distributions on <math>M,</math>
* the space <math>\mathcal{O}(M)</math> of holomorphic functions on arbitrary complex manifold <math>M,</math> and its strong dual space <math>\mathcal{O}^{\prime}(M)</math> of analytic functionals on <math>M,</math>
* the [[Schwartz space]] <math>\mathcal{S}\left(\R^n\right)</math> on <math>\R^n,</math> and its strong dual space <math>\mathcal{S}^{\prime}\left(\R^n\right)</math> of tempered distributions on <math>\R^n.</math>
</li>
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