# 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.

## Definition

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

$\displaystyle{ |y|_B = \sup_{x \in B}|\langle x, y\rangle|. }$

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

$\displaystyle{ |y|_B = \sup_{x\in B}|\langle x, y\rangle| \lt \infty. }$

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

$\displaystyle{ |y|_B = \sup_{x \in B}|\langle x, y\rangle|,\qquad y \in Y,\qquad B \in \mathcal{B}. }$

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

$\displaystyle{ |f|_B = \sup_{x \in B} | f(x) |,\qquad f \in X', }$

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

## Examples

• If X is a normed vector space, then its (continuous) dual space $\displaystyle{ X' }$ with the strong topology coincides with the Banach dual space $\displaystyle{ X' }$, i.e. with the space $\displaystyle{ X' }$ with the topology induced by the operator norm. Conversely $\displaystyle{ \beta(X, X') }$-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 $\displaystyle{ \beta(X, X') }$ on $\displaystyle{ X }$ and with the Mackey topology on X generated by the pairing $\displaystyle{ (X, X') }$.