Strong topology (polar topology)

Short description: Dual space topology of uniform convergence on bounded subsets

In functional analysis and related areas of mathematics the strong topology on the continuous dual space of a topological vector space (TVS) X is the finest polar topology, the topology with the most open sets, on a dual pair. The coarsest polar topology is called weak topology. When the continuous dual space of a TVS X is endowed with this topology then it is called the strong dual space of X.

## Contents

* 1 Definition
* 2 Examples
* 3 Properties
* 4 See also
* 5 References

## Definition

Let [math]\displaystyle{ (X, Y, \langle \cdot, \cdot \rangle) }[/math] be a dual pair of vector spaces over the field [math]\displaystyle{ {\mathbb F} }[/math] of real numbers [math]\displaystyle{ \R }[/math] or complex numbers [math]\displaystyle{ \C. }[/math] For any [math]\displaystyle{ B \subseteq X }[/math] and any [math]\displaystyle{ y \in Y, }[/math] define

[math]\displaystyle{ |y|_B = \sup_{x \in B}|\langle x, y\rangle|. }[/math]

A subset [math]\displaystyle{ B \subseteq X }[/math] is said to be bounded by a subset [math]\displaystyle{ C \subseteq Y }[/math] if [math]\displaystyle{ |y|_B \lt \infty }[/math] for all [math]\displaystyle{ y \in C. }[/math] Let [math]\displaystyle{ \mathcal{B} }[/math] denote the family of all subsets [math]\displaystyle{ B \subseteq X }[/math] bounded by elements of [math]\displaystyle{ Y }[/math]; that is, [math]\displaystyle{ \mathcal{B} }[/math] is the set of all subsets [math]\displaystyle{ B \subseteq X }[/math] such that for every [math]\displaystyle{ y \in Y, }[/math]

[math]\displaystyle{ |y|_B = \sup_{x\in B}|\langle x, y\rangle| \lt \infty. }[/math]

Then the strong topology [math]\displaystyle{ \beta(Y, X, \langle \cdot, \cdot \rangle) }[/math] on [math]\displaystyle{ Y, }[/math] also denoted by [math]\displaystyle{ b(Y, X, \langle \cdot, \cdot \rangle) }[/math] or simply [math]\displaystyle{ \beta(Y, X) }[/math] or [math]\displaystyle{ b(Y, X) }[/math] if the pairing [math]\displaystyle{ \langle \cdot, \cdot\rangle }[/math] is understood, is defined as the locally convex topology on [math]\displaystyle{ Y }[/math] generated by the seminorms of the form

[math]\displaystyle{ |y|_B = \sup_{x \in B}|\langle x, y\rangle|,\qquad y \in Y,\qquad B \in \mathcal{B}. }[/math]

In the special case when X is a locally convex space, the strong topology on the (continuous) dual space [math]\displaystyle{ X' }[/math] (i.e. on the space of all continuous linear functionals [math]\displaystyle{ f : X \to {\mathbb F} }[/math]) is defined as the strong topology [math]\displaystyle{ \beta(X', X) }[/math], and it coincides with the topology of uniform convergence on bounded sets in [math]\displaystyle{ X, }[/math] i.e. with the topology on [math]\displaystyle{ X' }[/math] generated by the seminorms of the form

[math]\displaystyle{ |f|_B = \sup_{x \in B} | f(x) |,\qquad f \in X', }[/math]

where [math]\displaystyle{ B }[/math] runs over the family of all bounded sets in [math]\displaystyle{ X. }[/math] The space [math]\displaystyle{ X' }[/math] with this topology is called strong dual space of the space [math]\displaystyle{ X }[/math] and is denoted by [math]\displaystyle{ X'_{\beta}. }[/math]

## Examples

* If X is a normed vector space, then its (continuous) dual space [math]\displaystyle{ X' }[/math] with the strong topology coincides with the Banach dual space [math]\displaystyle{ X' }[/math], i.e. with the space [math]\displaystyle{ X' }[/math] with the topology induced by the operator norm. Conversely [math]\displaystyle{ \beta(X, X') }[/math]-topology on X is identical to the topology induced by the norm on X.

## Properties

* If X is a barrelled space, then its topology coincides with the strong topology [math]\displaystyle{ \beta(X, X') }[/math] on [math]\displaystyle{ X }[/math] and with the Mackey topology on X generated by the pairing [math]\displaystyle{ (X, X') }[/math].

## See also

* Dual topology
* Dual system
* Reflexive space
* Polar topology - Dual space topology of uniform convergence on some sub-collection of bounded subsets
* Semi-reflexive space
* Strong dual space - Continuous dual space endowed with the topology of uniform convergence on bounded sets
* Topologies on spaces of linear maps

## References

* Schaefer, Helmuth H. (1966). Topological vector spaces. New York: The MacMillan Company. ISBN 0-387-98726-6.

* v
* t
* e

Functional analysis (topics)

Topological vector spaces|

* Asplund
* Banach (list)
* Banach lattice
* Barrelled
* Bornological
* Brauner
* F-space
* Fréchet (tame)
* Hilbert (Inner product space
* Polarization identity)
* LF-space
* Locally convex (Seminorms/Minkowski functionals)
* Mackey
* Montel
* Nuclear
* Normed (norm)
* Quasinormed
* Reflexive
* Riesz
* Smith
* Stereotype
* Strictly convex
* Webbed
* Topological tensor product (of Hilbert spaces)

Topologies of function spaces|

* Dual
* Dual space (Dual norm)
* Operator
* Ultraweak
* Weak (polar
* operator)
* Mackey
* Strong (polar
* operator)
* Ultrastrong
* Uniform convergence

Linear operators|

* Adjoint
* Bilinear (form
* operator
* sesquilinear)
* (Un)Bounded
* Closed
* Compact (on Hilbert spaces)
* (Dis)Continuous
* Densely defined
* Fredholm
* Hilbert–Schmidt
* Functionals (positive)
* Normal
* Nuclear
* Self-adjoint
* Strictly singular
* Trace class
* Transpose
* Unitary

Operator theory|

* Banach algebras
* C*-algebras
* Spectrum (C*-algebra
* radius)
* Spectral theory (of ODEs
* Spectral theorem)
* Polar decomposition
* Singular value decomposition

Theorems|

* Banach–Alaoglu
* Banach–Mazur
* Banach–Saks
* Banach–Schauder (open mapping)
* Banach–Steinhaus (Uniform boundedness)
* Bessel's inequality
* Cauchy–Schwarz inequality
* Closed graph
* Closed range
* Eberlein–Šmulian
* Freudenthal spectral
* Gelfand–Mazur
* Gelfand–Naimark
* Goldstine
* Hahn–Banach (hyperplane separation)
* Kakutani fixed-point
* Krein–Milman
* Lomonosov's invariant subspace
* Mackey–Arens
* Mazur's lemma
* M. Riesz extension
* Riesz representation
* Parseval's identity
* Schauder fixed-point

Analysis|

* Abstract Wiener space
* Bochner space
* Differentiation in Fréchet spaces
* Derivatives (Fréchet
* Gateaux
* functional
* holomorphic)
* Integrals (Bochner
* Dunford
* Gelfand–Pettis
* regulated
* Paley–Wiener
* weak)
* Functional calculus (Borel
* continuous
* holomorphic)
* Inverse function theorem (Nash–Moser theorem)
* Measures (Lebesgue
* Projection-valued
* Vector)
* Weakly measurable function

Types of sets|

* Absolutely convex
* Absorbing
* Balanced
* Bounded
* Convex
* Convex cone (subset)
* Linear cone (subset)
* Radial
* Star-shaped
* Symmetric
* Zonotope

Subsets / set operations|

* Algebraic interior (core)
* Bounding points
* Convex hull
* Extreme point
* Interior
* Minkowski addition
* Polar

* v
* t
* e

Duality and spaces of linear maps

Basic concepts|

* Dual space
* Dual system
* Dual topology
* Duality
* Polar set
* Polar topology
* Topologies on spaces of linear maps

Topologies|

* Dual norm
* Ultraweak/Weak-*
* Weak (polar
* operator)
* Mackey
* Strong dual (polar topology
* operator)
* Ultrastrong

Main results|

* Banach–Alaoglu
* Mackey–Arens

Maps|

* Transpose of a linear map

Subsets|

* Saturated family

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