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

$|y{|}_B=\underset{x\in B}{\sup }|⟨x,y⟩|.$

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

$|y{|}_B=\underset{x\in B}{\sup }|⟨x,y⟩|<\mathrm{\infty }.$

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

$|y{|}_B=\underset{x\in B}{\sup }|⟨x,y⟩|,y\in Y,B\in {ℬ}.$

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

$|f{|}_B=\underset{x\in B}{\sup }|f(x)|,f\in X^{\prime },$

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

Examples

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

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.