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

Given a dual pair $(X,Y,\langle , \rangle)$ the strong topology $\beta(Y, X)$ on $Y$ is the polar topology defined by using the family of all sets in $X$ where the polar set in $Y$ is absorbent.

Examples

• Given a normed vector space $X$ and its continuous dual $X'$ then $\beta(X', X)$-topology on $X'$ is identical to the topology induced by the operator norm. Conversely $\beta(X, X')$-topology on $X$ is identical to the topology induced by the norm.

Properties

• In barrelled spaces the strong topology is identical to the Mackey topology.