In the area of mathematics known asfunctional analysis, asemi-reflexive space is alocally convextopological vector space (TVS)X such that the canonical evaluation map fromX into its bidual (which is thestrong dual ofX) is bijective.If this map is also an isomorphism of TVSs then it is calledreflexive.

Semi-reflexive spaces play an important role in the general theory oflocally convex TVSs.Since anormable TVS is semi-reflexive if and only if it is reflexive, the concept of semi-reflexivity is primarily used with TVSs that are not normable.

Suppose thatX is atopological vector space (TVS) over the field${\displaystyle \mathbb{F}}$ (which is either the real or complex numbers) whosecontinuous dual space,${\displaystyle X^{\mathrm{\prime }}}$,separates points onX (i.e. for any${\displaystyle x\in X}$ there exists some${\displaystyle x^{\mathrm{\prime }}\in X^{\mathrm{\prime }}}$ such that${\displaystyle x^{\mathrm{\prime }}(x)\ne 0}$).Let${\displaystyle X_b^{\mathrm{\prime }}}$ and${\displaystyle X_{\beta }^{\mathrm{\prime }}}$ both denote thestrong dual ofX, which is the vector space${\displaystyle X^{\mathrm{\prime }}}$ of continuous linear functionals onX endowed with thetopology of uniform convergence onbounded subsets ofX;this topology is also called thestrong dual topology and it is the "default" topology placed on a continuous dual space (unless another topology is specified).IfX is a normed space, then the strong dual ofX is the continuous dual space${\displaystyle X^{\mathrm{\prime }}}$ with its usual norm topology.Thebidual ofX, denoted by${\displaystyle X^{\mathrm{\prime }\mathrm{\prime }}}$, is the strong dual of${\displaystyle X_b^{\mathrm{\prime }}}$; that is, it is the space${\displaystyle {\left(X_b^{\mathrm{\prime }}\right)}_b^{\mathrm{\prime }}}$.^[1]

For any${\displaystyle x\in X,}$ let${\displaystyle J_x:X^{\mathrm{\prime }}\to \mathbb{F}}$ be defined by${\displaystyle J_x\left(x^{\mathrm{\prime }}\right)=x^{\mathrm{\prime }}(x)}$, where${\displaystyle J_x}$ is called theevaluation map atx;since${\displaystyle J_x:X_b^{\mathrm{\prime }}\to \mathbb{F}}$ is necessarily continuous, it follows that${\displaystyle J_x\in {\left(X_b^{\mathrm{\prime }}\right)}^{\mathrm{\prime }}}$.Since${\displaystyle X^{\mathrm{\prime }}}$ separates points onX, the map${\displaystyle J:X\to {\left(X_b^{\mathrm{\prime }}\right)}^{\mathrm{\prime }}}$ defined by${\displaystyle J(x):=J_x}$ is injective where this map is called theevaluation map or thecanonical map.This map was introduced byHans Hahn in 1927.^[2]

We callXsemireflexive if${\displaystyle J:X\to {\left(X_b^{\mathrm{\prime }}\right)}^{\mathrm{\prime }}}$ is bijective (or equivalently,surjective) and we callXreflexive if in addition${\displaystyle J:X\to X^{\mathrm{\prime }\mathrm{\prime }}={\left(X_b^{\mathrm{\prime }}\right)}_b^{\mathrm{\prime }}}$ is an isomorphism of TVSs.^[1]IfX is a normed space thenJ is a TVS-embedding as well as anisometry onto its range;furthermore, byGoldstine's theorem (proved in 1938), the range ofJ is a dense subset of the bidual${\displaystyle \left(X^{\mathrm{\prime }\mathrm{\prime }},\sigma \left(X^{\mathrm{\prime }\mathrm{\prime }},X^{\mathrm{\prime }}\right)\right)}$.^[2]Anormable space is reflexive if and only if it is semi-reflexive.ABanach space is reflexive if and only if its closed unit ball is${\displaystyle \sigma \left(X^{\mathrm{\prime }},X\right)}$-compact.^[2]

LetX be a topological vector space over a number field${\displaystyle \mathbb{F}}$ (ofreal numbers${\displaystyle \mathbb{R}}$ orcomplex numbers${\displaystyle \mathbb{C}}$).Consider itsstrong dual space${\displaystyle X_b^{\mathrm{\prime }}}$, which consists of allcontinuouslinear functionals${\displaystyle f:X\to \mathbb{F}}$ and is equipped with thestrong topology${\displaystyle b\left(X^{\mathrm{\prime }},X\right)}$, that is, the topology of uniform convergence on bounded subsets inX.The space${\displaystyle X_b^{\mathrm{\prime }}}$ is a topological vector space (to be more precise, a locally convex space), so one can consider its strong dual space${\displaystyle {\left(X_b^{\mathrm{\prime }}\right)}_b^{\mathrm{\prime }}}$, which is called thestrong bidual space forX.It consists of allcontinuous linear functionals${\displaystyle h:X_b^{\mathrm{\prime }}\to \mathbb{F}}$ and is equipped with the strong topology${\displaystyle b\left({\left(X_b^{\mathrm{\prime }}\right)}^{\mathrm{\prime }},X_b^{\mathrm{\prime }}\right)}$.Each vector${\displaystyle x\in X}$ generates a map${\displaystyle J(x):X_b^{\mathrm{\prime }}\to \mathbb{F}}$ by the following formula:

$${\displaystyle J(x)(f)=f(x),f\in X^{\prime }.}$$

This is a continuous linear functional on${\displaystyle X_b^{\mathrm{\prime }}}$, that is,${\displaystyle J(x)\in {\left(X_b^{\mathrm{\prime }}\right)}_b^{\mathrm{\prime }}}$.One obtains a map called theevaluation map or thecanonical injection:

$${\displaystyle J:X\to {\left(X_b^{\mathrm{\prime }}\right)}_b^{\mathrm{\prime }}.}$$

which is a linear map.IfX is locally convex, from theHahn–Banach theorem it follows thatJ is injective and open (that is, for each neighbourhood of zero${\displaystyle U}$ inX there is a neighbourhood of zeroV in${\displaystyle {\left(X_b^{\mathrm{\prime }}\right)}_b^{\mathrm{\prime }}}$ 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 calledsemi-reflexive if the evaluation map${\displaystyle J:X\to {\left(X_b^{\mathrm{\prime }}\right)}_b^{\mathrm{\prime }}}$ is surjective (hence bijective); it is calledreflexive if the evaluation map${\displaystyle J:X\to {\left(X_b^{\mathrm{\prime }}\right)}_b^{\mathrm{\prime }}}$ is surjective and continuous, in which caseJ will be anisomorphism of TVSs).

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

1. X is semireflexive;
2. the weak topology onX had the Heine-Borel property (that is, for the weak topology${\displaystyle \sigma \left(X,X^{\mathrm{\prime }}\right)}$, every closed and bounded subset of${\displaystyle X_{\sigma }}$ is weakly compact).^[1]
3. If linear form on${\displaystyle X^{\mathrm{\prime }}}$ that continuous when${\displaystyle X^{\mathrm{\prime }}}$ has the strong dual topology, then it is continuous when${\displaystyle X^{\mathrm{\prime }}}$ has the weak topology;^[3]
4. ${\displaystyle X_{\tau }^{\mathrm{\prime }}}$ isbarrelled, where the${\displaystyle \tau }$ indicates theMackey topology on${\displaystyle X^{\mathrm{\prime }}}$;^[3]
5. X weak the weak topology${\displaystyle \sigma \left(X,X^{\mathrm{\prime }}\right)}$ isquasi-complete.^[3]

Everysemi-Montel space is semi-reflexive and everyMontel 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 isbarrelled. Every semi-reflexive space isquasi-complete.^[3] Every semi-reflexive normed space is a reflexive Banach space.^[6] The strong dual of a semireflexive space is barrelled.^[7]

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

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

IfX is anormed space then the following are equivalent:

1. X is reflexive;
2. the closed unit ball is compact whenX has the weak topology${\displaystyle \sigma \left(X,X^{\mathrm{\prime }}\right)}$.^[9]
3. X is a Banach space and${\displaystyle X_b^{\mathrm{\prime }}}$ is reflexive.^[10]

Every non-reflexive infinite-dimensionalBanach space is adistinguished 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-reflexivecountably barrelled space that is notbarrelled.^[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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• Banach spaces
• Duality theories