Reflexive space

Reflexive locally convex spaces

The notion of reflexive Banach space can be generalized to topological vector spaces in the following way.

Let $X$ be a topological vector space over a number field $\mathbb{𝔽}$ (of real numbers $\mathbb{ℝ}$ or complex numbers $\mathbb{ℂ}$). Consider its strong dual space $X_b^{\prime },$ which consists of all continuous linear functionals $f:X\to \mathbb{𝔽}$ and is equipped with the strong topology $b(X^{\prime },X),$ that is,, the topology of uniform convergence on bounded subsets in $X.$ The space $X_b^{\prime }$ is a topological vector space (to be more precise, a locally convex space), so one can consider its strong dual space ${\left(X_b^{\prime }\right)}_b^{\prime },$ which is called the strong bidual space for $X.$ It consists of all continuous linear functionals $h:X_b^{\prime }\to \mathbb{𝔽}$ and is equipped with the strong topology $b({\left(X_b^{\prime }\right)}^{\prime },X_b^{\prime }).$ Each vector $x\in X$ generates a map $J(x):X_b^{\prime }\to \mathbb{𝔽}$ by the following formula: $$J(x)(f)=f(x),f\in X^{\prime }.$$ This is a continuous linear functional on $X_b^{\prime },$ that is,, $J(x)\in {\left(X_b^{\prime }\right)}_b^{\prime }.$ This induces a map called the evaluation map: $$J:X\to {\left(X_b^{\prime }\right)}_b^{\prime }.$$ This map is linear. If $X$ is locally convex, from the Hahn–Banach theorem it follows that $J$ is injective and open (that is, for each neighbourhood of zero $U$ in $X$ there is a neighbourhood of zero $V$ in ${\left(X_b^{\prime }\right)}_b^{\prime }$ such that $J(U)\supseteq V\cap J(X)$). But it can be non-surjective and/or discontinuous.

A locally convex space $X$ is called

• semi-reflexive if the evaluation map $J:X\to {\left(X_b^{\prime }\right)}_b^{\prime }$ is surjective (hence bijective),
• reflexive if the evaluation map $J:X\to {\left(X_b^{\prime }\right)}_b^{\prime }$ is surjective and continuous (in this case $J$ is an isomorphism of topological vector spaces).

Semireflexive spaces

Main article: Semi-reflexive space

Characterizations

If $X$ is a Hausdorff locally convex space then the following are equivalent:

1. $X$ is semireflexive;
2. The weak topology on $X$ had the Heine-Borel property (that is, for the weak topology $\sigma (X,X^{\prime }),$ every closed and bounded subset of $X_{\sigma }$ is weakly compact).
3. If linear form on $X^{\prime }$ that continuous when $X^{\prime }$ has the strong dual topology, then it is continuous when $X^{\prime }$ has the weak topology;
4. $X_{\tau }^{\prime }$ is barreled;
5. $X$ with the weak topology $\sigma (X,X^{\prime })$ is quasi-complete.

Characterizations of reflexive spaces

If $X$ is a Hausdorff locally convex space then the following are equivalent:

1. $X$ is reflexive;
2. $X$ is semireflexive and infrabarreled;
3. $X$ is semireflexive and barreled;
4. $X$ is barreled and the weak topology on $X$ had the Heine-Borel property (that is, for the weak topology $\sigma (X,X^{\prime }),$ every closed and bounded subset of $X_{\sigma }$ is weakly compact).
5. $X$ is semireflexive and quasibarrelled.

If $X$ is a normed space then the following are equivalent:

1. $X$ is reflexive;
2. The closed unit ball is compact when $X$ has the weak topology $\sigma (X,X^{\prime }).$
3. $X$ is a Banach space and $X_b^{\prime }$ is reflexive.
4. Every sequence ${\left(C_n\right)}_{n=1}^{\mathrm{\infty }},$ with $C_{n+1}\subseteq C_n$ for all $n$ of nonempty closed bounded convex subsets of $X$ has nonempty intersection.

Sufficient conditions

Normed spaces

A normed space that is semireflexive is a reflexive Banach space. A closed vector subspace of a reflexive Banach space is reflexive.

Let $X$ be a Banach space and $M$ a closed vector subspace of $X.$ If two of $X,M,$ and $X/M$ are reflexive then they all are. This is why reflexivity is referred to as a three-space property.

Topological vector spaces

If a barreled locally convex Hausdorff space is semireflexive then it is reflexive.

The strong dual of a reflexive space is reflexive.Every Montel space is reflexive. And the strong dual of a Montel space is a Montel space (and thus is reflexive).

Properties

A locally convex Hausdorff reflexive space is barrelled. If $X$ is a normed space then $I:X\to X^{\prime \prime }$ is an isometry onto a closed subspace of $X^{\prime \prime }.$ This isometry can be expressed by: $$\left|x\right|={\sup }_{\stackrel{x^{\prime }\in X^{\prime },}{\left|x^{\prime }\right|\le 1}}\left|⟨x^{\prime },x⟩\right|.$$

Suppose that $X$ is a normed space and $X^{\prime \prime }$ is its bidual equipped with the bidual norm. Then the unit ball of $X,$ $I(\{x\in X:\Vert x\Vert \le 1\})$ is dense in the unit ball $\{x^{\prime \prime }\in X^{\prime \prime }:\Vert x^{\prime \prime }\Vert \le 1\}$ of $X^{\prime \prime }$ for the weak topology $\sigma (X^{\prime \prime },X^{\prime }).$

Examples

1. Every finite-dimensional Hausdorff topological vector space is reflexive, because $J$ is bijective by linear algebra, and because there is a unique Hausdorff vector space topology on a finite dimensional vector space.
2. A normed space $X$ 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 $X$ its dual normed space $X^{\prime }$ coincides as a topological vector space with the strong dual space $X_b^{\prime }.$ As a corollary, the evaluation map $J:X\to X^{\prime \prime }$ coincides with the evaluation map $J:X\to {\left(X_b^{\prime }\right)}_b^{\prime },$ and the following conditions become equivalent:
   1. $X$ is a reflexive normed space (that is, $J:X\to X^{\prime \prime }$ is an isomorphism of normed spaces),
   2. $X$ is a reflexive locally convex space (that is, $J:X\to {\left(X_b^{\prime }\right)}_b^{\prime }$ is an isomorphism of topological vector spaces),
   3. $X$ is a semi-reflexive locally convex space (that is, $J:X\to {\left(X_b^{\prime }\right)}_b^{\prime }$ is surjective).
3. A (somewhat artificial) example of a semi-reflexive space that is not reflexive is obtained as follows: let $Y$ be an infinite dimensional reflexive Banach space, and let $X$ be the topological vector space $(Y,\sigma (Y,Y^{\prime })),$ that is, the vector space $Y$ equipped with the weak topology. Then the continuous dual of $X$ and $Y^{\prime }$ are the same set of functionals, and bounded subsets of $X$ (that is, weakly bounded subsets of $Y$) are norm-bounded, hence the Banach space $Y^{\prime }$ is the strong dual of $X.$ Since $Y$ is reflexive, the continuous dual of $X^{\prime }=Y^{\prime }$ is equal to the image $J(X)$ of $X$ under the canonical embedding $J,$ but the topology on $X$ (the weak topology of $Y$) is not the strong topology $\beta (X,X^{\prime }),$ that is equal to the norm topology of $Y.$
4. Montel spaces are reflexive locally convex topological vector spaces. In particular, the following functional spaces frequently used in functional analysis are reflexive locally convex spaces:
   • the space $C^{\mathrm{\infty }}(M)$ of smooth functions on arbitrary (real) smooth manifold $M,$ and its strong dual space ${\left(C^{\mathrm{\infty }}\right)}^{\prime }(M)$ of distributions with compact support on $M,$
   • the space ${𝒟}(M)$ of smooth functions with compact support on arbitrary (real) smooth manifold $M,$ and its strong dual space ${{𝒟}}^{\prime }(M)$ of distributions on $M,$
   • the space ${𝒪}(M)$ of holomorphic functions on arbitrary complex manifold $M,$ and its strong dual space ${{𝒪}}^{\prime }(M)$ of analytic functionals on $M,$
   • the Schwartz space ${𝒮}\left({\mathbb{ℝ}}^n\right)$ on ${\mathbb{ℝ}}^n,$ and its strong dual space ${{𝒮}}^{\prime }\left({\mathbb{ℝ}}^n\right)$ of tempered distributions on ${\mathbb{ℝ}}^n.$

Counter-examples

• There exists a non-reflexive locally convex TVS whose strong dual is reflexive.