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 $(X,X')$ .

References

Schaefer, Helmuth H. (1966). Topological vector spaces. New York: The MacMillan Company. ISBN 0-387-98726-6.

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Functional analysis (topics)
TVS types	Banach Banach Lattice Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert (pre-Hilbert Polarization identity) LF-space Locally convex (Seminorms/Minkowski functionals) Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Strictly convex Webbed Topological tensor product (of Hilbert spaces)
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