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 (X,Y,⟨⋅,⋅⟩) be a dual pair of vector spaces over the field 𝔽 of real numbers ℝ or complex numbers ℂ. For any B⊆X and any y∈Y, define

|y|B=supx∈B|⟨x,y⟩|.

A subset B⊆X is said to be bounded by a subset C⊆Y if |y|B<∞ for all y∈C. Let ℬ denote the family of all subsets B⊆X bounded by elements of Y; that is, ℬ is the set of all subsets B⊆X such that for every y∈Y,

|y|B=supx∈B|⟨x,y⟩|<∞.

Then the strong topology β(Y,X,⟨⋅,⋅⟩) on Y, also denoted by b(Y,X,⟨⋅,⋅⟩) or simply β(Y,X) or b(Y,X) if the pairing ⟨⋅,⋅⟩ is understood, is defined as the locally convex topology on Y generated by the seminorms of the form

|y|B=supx∈B|⟨x,y⟩|,y∈Y,B∈ℬ.

In the special case when X is a locally convex space, the strong topology on the (continuous) dual space X′ (i.e. on the space of all continuous linear functionals f:X->𝔽) is defined as the strong topology β(X′,X), and it coincides with the topology of uniform convergence on bounded sets in X, i.e. with the topology on X′ generated by the seminorms of the form

|f|B=supx∈B|f(x)|,f∈X′,

where B runs over the family of all bounded sets in X. The space X′ with this topology is called strong dual space of the space X and is denoted by X'β.

## Examples

* If X is a normed vector space, then its (continuous) dual space X′ with the strong topology coincides with the Banach dual space X′, i.e. with the space X′ with the topology induced by the operator norm. Conversely β(X,X′)-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 β(X,X′) on X and with the Mackey topology on X generated by the pairing (X,X′).

## See also

* Dual topology
* Dual system - Dual pair of vector spaces
* Reflexive space - Locally convex topological vector 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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