View source for Strong dual space

{{short description|Continuous dual space endowed with the topology of uniform convergence on bounded sets}}
In [[Functional analysis|functional analysis]] and related areas of [[Mathematics|mathematics]], the '''strong dual space''' of a [[Topological vector space|topological vector space]] (TVS) <math>X</math> is the continuous dual space <math>X^{\prime}</math> of <math>X</math> equipped with the '''strong''' ('''dual''') '''topology''' or the '''topology of uniform convergence on bounded subsets of <math>X,</math>''' where this topology is denoted by <math>b\left(X^{\prime}, X\right)</math> or <math>\beta\left(X^{\prime}, X\right).</math> The [[Comparison of topologies|coarsest]] polar topology is called [[Weak topology (polar topology)|weak topology]]. 
The strong dual space plays such an important role in modern functional analysis, that the continuous dual space is usually assumed to have the strong dual topology unless indicated otherwise. 
To emphasize that the continuous dual space, <math>X^{\prime},</math> has the strong dual topology, <math>X^{\prime}_b</math> or <math>X^{\prime}_{\beta}</math> may be written. 

== Strong dual topology ==

Throughout, all vector spaces will be assumed to be over the field <math>\mathbb{F}</math> of either the [[Real number|real number]]s <math>\R</math> or [[Complex number|complex number]]s <math>\C.</math> 

=== Definition from a dual system ===
{{Main|Dual system}}

Let <math>(X, Y, \langle \cdot, \cdot \rangle)</math> be a [[Dual pair|dual pair]] of vector spaces over the field <math>\mathbb{F}</math> of [[Real number|real number]]s <math>\R</math> or [[Complex number|complex number]]s <math>\C.</math> 
For any <math>B \subseteq X</math> and any <math>y \in Y,</math> define 
<math display="block">|y|_B = \sup_{x \in B}|\langle x, y\rangle|.</math>

Neither <math>X</math> nor <math>Y</math> has a topology so say a subset <math>B \subseteq X</math> is said to be '''{{em|bounded by a subset <math>C \subseteq Y</math>}}''' if <math>|y|_B < \infty</math> for all <math>y \in C.</math> 
So a subset <math>B \subseteq X</math> is called {{em|bounded}} if and only if 
<math display="block">\sup_{x \in B} |\langle x, y \rangle| < \infty \quad \text{ for all } y \in Y.</math> 
This is equivalent to the usual notion of [[Bounded set (topological vector space)|bounded subsets]] when <math>X</math> is given the weak topology induced by <math>Y,</math> which is a Hausdorff locally convex topology. 

Let <math>\mathcal{B}</math> denote the [[Family of sets|family]] of all subsets <math>B \subseteq X</math> bounded by elements of <math>Y</math>; that is, <math>\mathcal{B}</math> is the set of all subsets <math>B \subseteq X</math> such that for every <math>y \in Y,</math>
<math display="block">|y|_B = \sup_{x\in B}|\langle x, y\rangle| < \infty.</math>
Then the '''{{em|strong topology}}''' <math>\beta(Y, X, \langle \cdot, \cdot \rangle)</math> on <math>Y,</math> also denoted by <math>b(Y, X, \langle \cdot, \cdot \rangle)</math> or simply <math>\beta(Y, X)</math> or <math>b(Y, X)</math> if the pairing <math>\langle \cdot, \cdot\rangle</math> is understood, is defined as the [[Locally convex topological vector space|locally convex]] topology on <math>Y</math> generated by the seminorms of the form
<math display="block">|y|_B = \sup_{x \in B} |\langle x, y\rangle|,\qquad y \in Y, \qquad B \in \mathcal{B}.</math>

The definition of the strong dual topology now proceeds as in the case of a TVS. 
Note that if <math>X</math> is a TVS whose continuous dual space separates point on <math>X,</math> then <math>X</math> is part of a canonical dual system <math>\left(X, X^{\prime}, \langle \cdot , \cdot \rangle\right)</math> 
where <math>\left\langle x, x^{\prime} \right\rangle := x^{\prime}(x).</math>
In the special case when <math>X</math> is a locally convex space, the '''{{em|strong topology}}''' on the (continuous) dual space <math>X^{\prime}</math> (that is, on the space of all continuous linear functionals <math>f : X \to \mathbb{F}</math>) is defined as the strong topology <math>\beta\left(X^{\prime}, X\right),</math> and it coincides with the topology of uniform convergence on [[Bounded set|bounded set]]s in <math>X,</math> i.e. with the topology on <math>X^{\prime}</math> generated by the seminorms of the form
<math display="block">|f|_B = \sup_{x \in B} |f(x)|, \qquad \text{ where } f \in X^{\prime},</math>
where <math>B</math> runs over the family of all [[Bounded set (topological vector space)|bounded set]]s in <math>X.</math> 
The space <math>X^{\prime}</math> with this topology is called '''{{em|strong dual space}}''' of the space <math>X</math> and is denoted by <math>X^{\prime}_{\beta}.</math>

=== Definition on a TVS ===

Suppose that <math>X</math> is a [[Topological vector space|topological vector space]] (TVS) over the field <math>\mathbb{F}.</math> 
Let <math>\mathcal{B}</math> be any fundamental system of [[Bounded set (topological vector space)|bounded sets]] of <math>X</math>; 
that is, <math>\mathcal{B}</math> is a [[Family of sets|family]] of bounded subsets of <math>X</math> such that every bounded subset of <math>X</math> is a subset of some <math>B \in \mathcal{B}</math>; 
the set of all bounded subsets of <math>X</math> forms a fundamental system of bounded sets of <math>X.</math> 
A basis of closed neighborhoods of the origin in <math>X^{\prime}</math> is given by the [[Polar set|polars]]: 
<math display="block">B^{\circ} := \left\{ x^{\prime} \in X^{\prime} : \sup_{x \in B} \left|x^{\prime}(x)\right| \leq 1 \right\}</math>
as <math>B</math> ranges over <math>\mathcal{B}</math>). 
This is a locally convex topology that is given by the set of [[Seminorm|seminorm]]s on <math>X^{\prime}</math>:
<math>\left|x^{\prime}\right|_{B} := \sup_{x \in B} \left|x^{\prime}(x)\right|</math>
as <math>B</math> ranges over <math>\mathcal{B}.</math> 

If <math>X</math> is normable then so is <math>X^{\prime}_{b}</math> and <math>X^{\prime}_{b}</math> will in fact be a [[Banach space]]. 
If <math>X</math> is a normed space with norm <math>\| \cdot \|</math> then <math>X^{\prime}</math> has a canonical norm (the [[Operator norm|operator norm]]) given by <math>\left\| x^{\prime} \right\| := \sup_{\| x \| \leq 1} \left| x^{\prime}(x) \right|</math>; 
the topology that this norm induces on <math>X^{\prime}</math> is identical to the strong dual topology.

== Bidual ==


The '''bidual''' or '''second dual''' of a TVS <math>X,</math> often denoted by <math>X^{\prime \prime},</math> is the strong dual of the strong dual of <math>X</math>:
<math display="block>X^{\prime \prime} \,:=\, \left(X^{\prime}_b\right)^{\prime}</math>
where <math>X^{\prime}_b</math> denotes <math>X^{\prime}</math> endowed with the strong dual topology <math>b\left(X^{\prime}, X\right).</math> 
Unless indicated otherwise, the vector space <math>X^{\prime \prime}</math> is usually assumed to be endowed with the strong dual topology induced on it by <math>X^{\prime}_b,</math> in which case it is called the '''strong bidual''' of <math>X</math>; that is, 
<math display="block>X^{\prime \prime} \,:=\, \left(X^{\prime}_b\right)^{\prime}_b</math>
where the vector space <math>X^{\prime \prime}</math> is endowed with the strong dual topology <math>b\left(X^{\prime\prime}, X^{\prime}_b\right).</math>

== Properties ==

Let <math>X</math> be a [[Locally convex topological vector space|locally convex]] TVS. 

* A convex [[Balanced set|balanced]] weakly compact subset of <math>X^{\prime}</math> is bounded in <math>X^{\prime}_b.</math>{{sfn|Schaefer|Wolff|1999|p=141}}
* Every weakly bounded subset of <math>X^{\prime}</math> is strongly bounded.{{sfn|Schaefer|Wolff|1999|p=142}}
* If <math>X</math> is a barreled space then <math>X</math>'s topology is identical to the strong dual topology <math>b\left(X, X^{\prime}\right)</math> and to the [[Mackey topology]] on <math>X.</math>
* If <math>X</math> is a metrizable locally convex space, then the strong dual of <math>X</math> is a [[Bornological space|bornological space]] if and only if it is an [[Infrabarreled space|infrabarreled space]], if and only if it is a barreled space.{{sfn|Schaefer|Wolff|1999|p=153}}
* If <math>X</math> is Hausdorff locally convex TVS then <math>\left(X, b\left(X, X^{\prime}\right)\right)</math> is [[Metrizable topological vector space|metrizable]] if and only if there exists a countable set <math>\mathcal{B}</math> of bounded subsets of <math>X</math> such that every bounded subset of <math>X</math> is contained in some element of <math>\mathcal{B}.</math>{{sfn|Narici|Beckenstein|2011|pp=225-273}}
* If <math>X</math> is locally convex, then this topology is finer than all other [[Topologies on spaces of linear maps|<math>\mathcal{G}</math>-topologies]] on <math>X^{\prime}</math> when considering only <math>\mathcal{G}</math>'s whose sets are subsets of <math>X.</math>
* If <math>X</math> is a [[Bornological space|bornological space]] (e.g. metrizable or [[LF-space]]) then <math>X^{\prime}_{b(X^{\prime}, X)}</math> is [[Complete topological vector space|complete]].

If <math>X</math> is a [[Barrelled space|barrelled space]], then its topology coincides with the strong topology <math>\beta\left(X, X^{\prime}\right)</math> on <math>X</math> and with the [[Mackey topology]] on  generated by the pairing <math>\left(X, X^{\prime}\right).</math>

== Examples ==

If <math>X</math> is a [[Normed vector space|normed vector space]], then its (continuous) dual space <math>X^{\prime}</math> with the strong topology coincides with the [[Banach space#dual space|Banach dual space]] <math>X^{\prime}</math>; that is, with the space <math>X^{\prime}</math> with the topology induced by the [[Operator norm|operator norm]]. Conversely <math>\left(X, X^{\prime}\right).</math>-topology on <math>X</math> is identical to the topology induced by the [[Norm (mathematics)|norm]] on <math>X.</math> 

== See also ==

* {{annotated link|Dual topology}}
* {{annotated link|Dual system}}
* {{annotated link|List of topologies}}
* {{annotated link|Polar topology}}
* {{annotated link|Reflexive space}}
* {{annotated link|Semi-reflexive space}}
* {{annotated link|Strong topology}}
* {{annotated link|Topologies on spaces of linear maps}}

== References ==
{{reflist}}

== Bibliography ==

* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|p=}} -->
* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn|Rudin |1991|p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|Wolff|1999|p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Treves|2006|p=}} -->
* {{cite book|author=Wong|title=Schwartz spaces, nuclear spaces, and tensor products|publisher=Springer-Verlag|publication-place=Berlin New York|year=1979|isbn=3-540-09513-6|oclc=5126158 }}

{{Functional Analysis}}
{{DualityInLCTVSs}}

[[Category:Functional analysis]]

[[Category:Topology of function spaces]]

{{Sourceattribution|Strong dual space}}