Strong topology (polar topology)

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.

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. 

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