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

${\displaystyle \forall y\in Y\qquad \sup _{x\in B}|\langle x,y\rangle |<\infty .}$

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

${\displaystyle ||y||_{B}=\sup _{x\in B}|\langle x,y\rangle |,\qquad y\in Y,\qquad B\in {\mathcal {B}}.}$

In the special case when ${\displaystyle X}$ is a locally convex space, the strong topology on the (continuous) dual space ${\displaystyle X'}$ (i.e. on the space of all continuous linear functionals ${\displaystyle f:X\to {\mathbb {F} }}$) is defined as the strong topology ${\displaystyle \beta (X',X)}$, and it coincides with the topology of uniform convergence on bounded sets in ${\displaystyle X}$, i.e. with the topology on ${\displaystyle X'}$ generated by the seminorms of the form

${\displaystyle ||f||_{B}=\sup _{x\in B}|f(x)|,\qquad f\in X',}$

where ${\displaystyle B}$ runs over the family of all bounded sets in ${\displaystyle X}$. The space ${\displaystyle X'}$ with this topology is called strong dual space of the space ${\displaystyle X}$ and is denoted by ${\displaystyle X'_{\beta }}$.

Examples

• If ${\displaystyle X}$ is a normed vector space, then its (continuous) dual space ${\displaystyle X'}$ with the strong topology coincides with the Banach dual space ${\displaystyle X'}$, i.e. with the space ${\displaystyle X'}$ with the topology induced by the operator norm. Conversely ${\displaystyle \beta (X,X')}$-topology on ${\displaystyle X}$ is identical to the topology induced by the norm on ${\displaystyle X}$.

Properties

• If ${\displaystyle X}$ is a barrelled space, then its topology coincides with the strong topology ${\displaystyle \beta (X,X')}$ on ${\displaystyle X}$ and with the Mackey topology on ${\displaystyle X}$ generated by the pairing ${\displaystyle (X,X')}$.

See also

• Reflexive space
• Semi-reflexive space
• Strong dual space

References

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