Strong topology (polar topology)

In functional analysis and related areas of mathematics the strong topology is the finest polar topology, the topology with the most open sets, on a dual pair. The coarsest polar topology is called weak topology.

Definition

Let (X,Y,\langle , \rangle) be a dual pair of vector spaces over the field {\mathbb F} of real ({\mathbb R}) or complex ({\mathbb C}) numbers. Let us denote by {\mathcal B} the system of all subsets B\subseteq X bounded by elements of Y in the following sense:

\forall y\in Y \qquad \sup_{x\in B}|\langle x, y\rangle|<\infty.

Then the strong topology \beta(Y,X) on Y is defined as the locally convex topology on Y generated by the seminorms of the form

||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 X' (i.e. on the space of all continuous linear functionals f:X\to{\mathbb F}) is defined as the strong topology \beta(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=\sup_{x\in B}|f(x)|,\qquad f\in 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'_\beta.

Examples

Properties

References