semi-reflexive space

Suppose that {{mvar|X}} is a topological vector space (TVS) over the field $\backslash mathbb\{F\}$ (which is either the real or complex numbers) whose continuous dual space, $X^\{\backslash prime\}$, separates points on {{mvar|X}} (i.e. for any $x\; \backslash in\; X$ there exists some $x^\{\backslash prime\}\; \backslash in\; X^\{\backslash prime\}$ such that $x^\{\backslash prime\}(x)\; \backslash neq\; 0$).

Let $X^\{\backslash prime\}\_b$ and $X^\{\backslash prime\}\_\{\backslash beta\}$ both denote the strong dual of {{mvar|X}}, which is the vector space $X^\{\backslash prime\}$ of continuous linear functionals on {{mvar|X}} endowed with the topology of uniform convergence on bounded subsets of {{mvar|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 {{mvar|X}} is a normed space, then the strong dual of {{mvar|X}} is the continuous dual space $X^\{\backslash prime\}$ with its usual norm topology.

The bidual of {{mvar|X}}, denoted by $X^\{\backslash prime\backslash prime\}$, is the strong dual of $X^\{\backslash prime\}\_b$; that is, it is the space $\backslash left(X^\{\backslash prime\}\_b\backslash right)^\{\backslash prime\}\_\{b\}$.{{sfn|Trèves|2006|pp=372–374}}

For any $x\; \backslash in\; X,$ let $J\_x\; :\; X^\{\backslash prime\}\; \backslash to\; \backslash mathbb\{F\}$ be defined by $J\_x\backslash left(x^\{\backslash prime\}\backslash right)\; =\; x^\{\backslash prime\}(x)$, where $J\_x$ is called the evaluation map at {{mvar|x}};

since $J\_x\; :\; X^\{\backslash prime\}\_b\; \backslash to\; \backslash mathbb\{F\}$ is necessarily continuous, it follows that $J\_x\; \backslash in\; \backslash left(X^\{\backslash prime\}\_b\backslash right)^\{\backslash prime\}$.

Since $X^\{\backslash prime\}$ separates points on {{mvar|X}}, the map $J\; :\; X\; \backslash to\; \backslash left(X^\{\backslash prime\}\_b\backslash right)^\{\backslash prime\}$ defined by $J(x)\; :=\; J\_x$ is injective where this map is called the evaluation map or the canonical map.

This map was introduced by Hans Hahn in 1927.{{sfn|Narici|Beckenstein|2011|pp=225–273}}

We call {{mvar|X}} semireflexive if $J\; :\; X\; \backslash to\; \backslash left(X^\{\backslash prime\}\_b\backslash right)^\{\backslash prime\}$ is bijective (or equivalently, surjective) and we call {{mvar|X}} reflexive if in addition $J\; :\; X\; \backslash to\; X^\{\backslash prime\backslash prime\}\; =\; \backslash left(X^\{\backslash prime\}\_b\backslash right)^\{\backslash prime\}\_b$ is an isomorphism of TVSs.{{sfn|Trèves|2006|pp=372–374}}

If {{mvar|X}} is a normed space then {{mvar|J}} is a TVS-embedding as well as an isometry onto its range;

furthermore, by Goldstine's theorem (proved in 1938), the range of {{mvar|J}} is a dense subset of the bidual $\backslash left(X^\{\backslash prime\backslash prime\},\; \backslash sigma\backslash left(X^\{\backslash prime\backslash prime\},\; X^\{\backslash prime\}\backslash right)\backslash right)$.{{sfn|Narici|Beckenstein|2011|pp=225–273}}

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 $\backslash sigma\backslash left(X^\{\backslash prime\},\; X\backslash right)$-compact.{{sfn|Narici|Beckenstein|2011|pp=225–273}}

Let {{mvar|X}} be a topological vector space over a number field $\backslash mathbb\{F\}$ (of real numbers $\backslash R$ or complex numbers $\backslash C$).

Consider its strong dual space $X^\{\backslash prime\}\_b$, which consists of all continuous linear functionals $f\; :\; X\; \backslash to\; \backslash mathbb\{F\}$ and is equipped with the strong topology $b\backslash left(X^\{\backslash prime\},\; X\backslash right)$, that is, the topology of uniform convergence on bounded subsets in {{mvar|X}}.

The space $X^\{\backslash prime\}\_b$ is a topological vector space (to be more precise, a locally convex space), so one can consider its strong dual space $\backslash left(X^\{\backslash prime\}\_b\backslash right)^\{\backslash prime\}\_\{b\}$, which is called the strong bidual space for {{mvar|X}}.

It consists of all

continuous linear functionals $h\; :\; X^\{\backslash prime\}\_b\; \backslash to\; \{\backslash mathbb\; F\}$ and is equipped with the strong topology $b\backslash left(\backslash left(X^\{\backslash prime\}\_b\backslash right)^\{\backslash prime\},\; X^\{\backslash prime\}\_b\; \backslash right)$.

Each vector $x\backslash in\; X$ generates a map $J(x)\; :\; X^\{\backslash prime\}\_b\; \backslash to\; \backslash mathbb\{F\}$ by the following formula:

$$J(x)(f)\; =\; f(x),\backslash qquad\; f\; \backslash in\; X\text{'}.$$

This is a continuous linear functional on $X^\{\backslash prime\}\_b$, that is, $J(x)\; \backslash in\; \backslash left(X^\{\backslash prime\}\_b\backslash right)^\{\backslash prime\}\_\{b\}$.

One obtains a map called the evaluation map or the canonical injection:

$$J\; :\; X\; \backslash to\; \backslash left(X^\{\backslash prime\}\_b\backslash right)^\{\backslash prime\}\_\{b\}.$$

which is a linear map.

If {{mvar|X}} is locally convex, from the Hahn–Banach theorem it follows that {{mvar|J}} is injective and open (that is, for each neighbourhood of zero $U$ in {{mvar|X}} there is a neighbourhood of zero {{mvar|V}} in $\backslash left(X^\{\backslash prime\}\_b\backslash right)^\{\backslash prime\}\_\{b\}$ such that $J(U)\; \backslash supseteq\; V\; \backslash cap\; J(X)$).

But it can be non-surjective and/or discontinuous.

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

If {{mvar|X}} is a Hausdorff locally convex space then the following are equivalent:

1. {{mvar|X}} is semireflexive;
2. the weak topology on {{mvar|X}} had the Heine-Borel property (that is, for the weak topology $\backslash sigma\backslash left(X,\; X^\{\backslash prime\}\backslash right)$, every closed and bounded subset of $X\_\{\backslash sigma\}$ is weakly compact).{{sfn|Trèves|2006|pp=372–374}}
3. If linear form on $X^\{\backslash prime\}$ that continuous when $X^\{\backslash prime\}$ has the strong dual topology, then it is continuous when $X^\{\backslash prime\}$ has the weak topology;{{sfn|Schaefer|Wolff|1999|p=144}}
4. $X^\{\backslash prime\}\_\{\backslash tau\}$ is barrelled, where the $\backslash tau$ indicates the Mackey topology on $X^\{\backslash prime\}$;{{sfn|Schaefer|Wolff|1999|p=144}}
5. {{mvar|X}} weak the weak topology $\backslash sigma\backslash left(X,\; X^\{\backslash prime\}\backslash right)$ is quasi-complete.{{sfn|Schaefer|Wolff|1999|p=144}}

{{Math theorem|name=Theorem{{sfn|Edwards|1965|loc=8.4.2}}|math_statement=

A locally convex Hausdorff space $X$ is semi-reflexive if and only if $X$ with the $\backslash sigma\backslash left(X,\; X^\{\backslash prime\}\backslash right)$-topology has the Heine–Borel property (i.e. weakly closed and bounded subsets of $X$ are weakly compact).

}}

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

If $X$ is a Hausdorff locally convex space then the canonical injection from $X$ into its bidual is a topological embedding if and only if $X$ is infrabarrelled.{{sfn|Narici|Beckenstein|2011|pp=488–491}}

The strong dual of a semireflexive space is barrelled.

Every semi-reflexive space is quasi-complete.{{sfn|Schaefer|Wolff|1999|p=144}}

Every semi-reflexive normed space is a reflexive Banach space.{{sfn|Schaefer|Wolff|1999|p=145}}

The strong dual of a semireflexive space is barrelled.{{sfn|Edwards|1965|loc=8.4.3}}

{{Main|Reflexive space}}

If {{mvar|X}} is a Hausdorff locally convex space then the following are equivalent:

1. {{mvar|X}} is reflexive;
2. {{mvar|X}} is semireflexive and barrelled;
3. {{mvar|X}} is barrelled and the weak topology on {{mvar|X}} had the Heine-Borel property (which means that for the weak topology $\backslash sigma\backslash left(X,\; X^\{\backslash prime\}\backslash right)$, every closed and bounded subset of $X\_\{\backslash sigma\}$ is weakly compact).{{sfn|Trèves|2006|pp=372-374}}
4. {{mvar|X}} is semireflexive and quasibarrelled.{{sfn|Khaleelulla|1982|pp=32–63}}

If {{mvar|X}} is a normed space then the following are equivalent:

1. {{mvar|X}} is reflexive;
2. the closed unit ball is compact when {{mvar|X}} has the weak topology $\backslash sigma\backslash left(X,\; X^\{\backslash prime\}\backslash right)$.{{sfn|Trèves|2006|p=376}}
3. {{mvar|X}} is a Banach space and $X^\{\backslash prime\}\_b$ is reflexive.{{sfn|Trèves|2006|p=377}}

Every non-reflexive infinite-dimensional Banach space is a distinguished space that is not semi-reflexive.{{sfn|Khaleelulla|1982|pp=28-63}}

If $X$ is a dense proper vector subspace of a reflexive Banach space then $X$ is a normed space that not semi-reflexive but its strong dual space is a reflexive Banach space.{{sfn|Khaleelulla|1982|pp=28-63}}

There exists a semi-reflexive countably barrelled space that is not barrelled.{{sfn|Khaleelulla|1982|pp=28-63}}

• 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

{{reflist|group=note}}

{{reflist}}

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{{Functional analysis}}

{{TopologicalVectorSpaces}}

Category:Banach spaces

Category:Duality theories