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

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