Brief definition edit

Suppose that X is a topological vector space (TVS) over the field ${\displaystyle \mathbb {F} }$  (which is either the real or complex numbers) whose continuous dual space, ${\displaystyle X^{\prime }}$ , separates points on X (i.e. for any ${\displaystyle x\in X}$  there exists some ${\displaystyle x^{\prime }\in X^{\prime }}$  such that ${\displaystyle x^{\prime }(x)\neq 0}$ ). Let ${\displaystyle X_{b}^{\prime }}$  and ${\displaystyle X_{\beta }^{\prime }}$  both denote the strong dual of X, which is the vector space ${\displaystyle X^{\prime }}$  of continuous linear functionals on X endowed with the topology of uniform convergence on bounded subsets of X; this topology is also called the strong dual topology and it is the "default" topology placed on a continuous dual space (unless another topology is specified). If X is a normed space, then the strong dual of X is the continuous dual space ${\displaystyle X^{\prime }}$  with its usual norm topology. The bidual of X, denoted by ${\displaystyle X^{\prime \prime }}$ , is the strong dual of ${\displaystyle X_{b}^{\prime }}$ ; that is, it is the space ${\displaystyle \left(X_{b}^{\prime }\right)_{b}^{\prime }}$ .^[1]

For any ${\displaystyle x\in X,}$  let ${\displaystyle J_{x}:X^{\prime }\to \mathbb {F} }$  be defined by ${\displaystyle J_{x}\left(x^{\prime }\right)=x^{\prime }(x)}$ , where ${\displaystyle J_{x}}$  is called the evaluation map at x; since ${\displaystyle J_{x}:X_{b}^{\prime }\to \mathbb {F} }$  is necessarily continuous, it follows that ${\displaystyle J_{x}\in \left(X_{b}^{\prime }\right)^{\prime }}$ . Since ${\displaystyle X^{\prime }}$  separates points on X, the map ${\displaystyle J:X\to \left(X_{b}^{\prime }\right)^{\prime }}$  defined by ${\displaystyle J(x):=J_{x}}$  is injective where this map is called the evaluation map or the canonical map. This map was introduced by Hans Hahn in 1927.^[2]

We call X semireflexive if ${\displaystyle J:X\to \left(X_{b}^{\prime }\right)^{\prime }}$  is bijective (or equivalently, surjective) and we call X reflexive if in addition ${\displaystyle J:X\to X^{\prime \prime }=\left(X_{b}^{\prime }\right)_{b}^{\prime }}$  is an isomorphism of TVSs.^[1] If X is a normed space then J is a TVS-embedding as well as an isometry onto its range; furthermore, by Goldstine's theorem (proved in 1938), the range of J is a dense subset of the bidual ${\displaystyle \left(X^{\prime \prime },\sigma \left(X^{\prime \prime },X^{\prime }\right)\right)}$ .^[2] A normable space is reflexive if and only if it is semi-reflexive. A Banach space is reflexive if and only if its closed unit ball is ${\displaystyle \sigma \left(X^{\prime },X\right)}$ -compact.^[2]

Detailed definition edit

Let X be a topological vector space over a number field ${\displaystyle \mathbb {F} }$  (of real numbers ${\displaystyle \mathbb {R} }$  or complex numbers ${\displaystyle \mathbb {C} }$ ). Consider its strong dual space ${\displaystyle X_{b}^{\prime }}$ , which consists of all continuous linear functionals ${\displaystyle f:X\to \mathbb {F} }$  and is equipped with the strong topology ${\displaystyle b\left(X^{\prime },X\right)}$ , that is, the topology of uniform convergence on bounded subsets in X. The space ${\displaystyle X_{b}^{\prime }}$  is a topological vector space (to be more precise, a locally convex space), so one can consider its strong dual space ${\displaystyle \left(X_{b}^{\prime }\right)_{b}^{\prime }}$ , which is called the strong bidual space for X. It consists of all continuous linear functionals ${\displaystyle h:X_{b}^{\prime }\to {\mathbb {F} }}$  and is equipped with the strong topology ${\displaystyle b\left(\left(X_{b}^{\prime }\right)^{\prime },X_{b}^{\prime }\right)}$ . Each vector ${\displaystyle x\in X}$  generates a map ${\displaystyle J(x):X_{b}^{\prime }\to \mathbb {F} }$  by the following formula:

This is a continuous linear functional on ${\displaystyle X_{b}^{\prime }}$ , that is, ${\displaystyle J(x)\in \left(X_{b}^{\prime }\right)_{b}^{\prime }}$ . One obtains a map called the evaluation map or the canonical injection:

which is a linear map. 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 ${\displaystyle U}$  in X there is a neighbourhood of zero V in ${\displaystyle \left(X_{b}^{\prime }\right)_{b}^{\prime }}$  such that ${\displaystyle J(U)\supseteq V\cap J(X)}$ ). But it can be non-surjective and/or discontinuous.

A locally convex space ${\displaystyle X}$  is called semi-reflexive if the evaluation map ${\displaystyle J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }}$  is surjective (hence bijective); it is called reflexive if the evaluation map ${\displaystyle J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }}$  is surjective and continuous, in which case J will be an isomorphism of TVSs).

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 ${\displaystyle \sigma \left(X,X^{\prime }\right)}$ , every closed and bounded subset of ${\displaystyle X_{\sigma }}$  is weakly compact).^[1]
3. If linear form on ${\displaystyle X^{\prime }}$  that continuous when ${\displaystyle X^{\prime }}$  has the strong dual topology, then it is continuous when ${\displaystyle X^{\prime }}$  has the weak topology;^[3]
4. ${\displaystyle X_{\tau }^{\prime }}$  is barrelled, where the ${\displaystyle \tau }$  indicates the Mackey topology on ${\displaystyle X^{\prime }}$ ;^[3]
5. X weak the weak topology ${\displaystyle \sigma \left(X,X^{\prime }\right)}$  is quasi-complete.^[3]

Every semi-Montel space is semi-reflexive and every Montel space is reflexive.

If ${\displaystyle X}$  is a Hausdorff locally convex space then the canonical injection from ${\displaystyle X}$  into its bidual is a topological embedding if and only if ${\displaystyle X}$  is infrabarrelled.^[5]

The strong dual of a semireflexive space is barrelled. Every semi-reflexive space is quasi-complete.^[3] Every semi-reflexive normed space is a reflexive Banach space.^[6] The strong dual of a semireflexive space is barrelled.^[7]

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

1. X is reflexive;
2. X is semireflexive and barrelled;
3. X is barrelled and the weak topology on X had the Heine-Borel property (which means that for the weak topology ${\displaystyle \sigma \left(X,X^{\prime }\right)}$ , every closed and bounded subset of ${\displaystyle X_{\sigma }}$  is weakly compact).^[1]
4. X is semireflexive and quasibarrelled.^[8]

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 ${\displaystyle \sigma \left(X,X^{\prime }\right)}$ .^[9]
3. X is a Banach space and ${\displaystyle X_{b}^{\prime }}$  is reflexive.^[10]

Every non-reflexive infinite-dimensional Banach space is a distinguished space that is not semi-reflexive.^[11] If ${\displaystyle X}$  is a dense proper vector subspace of a reflexive Banach space then ${\displaystyle X}$  is a normed space that not semi-reflexive but its strong dual space is a reflexive Banach space.^[11] There exists a semi-reflexive countably barrelled space that is not barrelled.^[11]

• Grothendieck space - A generalization which has some of the properties of reflexive spaces and includes many spaces of practical importance.
• Reflexive operator algebra
• Reflexive space

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