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.