Strong topology (polar topology)

In functional analysis and related areas of mathematics the strong topology on the continuous dual space of a topological vector space (TVS) X is the finest polar topology, the topology with the most open sets, on a dual pair. The coarsest polar topology is called weak topology. When the continuous dual space of a TVS X is endowed with this topology then it is called the strong dual space of X.

Definition

Let ${\displaystyle (X,Y,⟨\cdot ,\cdot ⟩)}$ be a dual pair of vector spaces over the field ${\displaystyle \mathbb{F}}$ of real numbers ${\displaystyle \text{R}}$ or complex numbers ${\displaystyle \text{C}.}$ For any ${\displaystyle B\subseteq X}$ and any ${\displaystyle y\in Y,}$ define

${\displaystyle |y{|}_B=\underset{x\in B}{sup}|⟨x,y⟩|.}$

A subset ${\displaystyle B\subseteq X}$ is said to be bounded by a subset ${\displaystyle C\subseteq Y}$ if ${\displaystyle |y{|}_B<\mathrm{\infty }}$ for all ${\displaystyle y\in C.}$ Let ${\displaystyle \mathcal{B}}$ denote the family of all subsets ${\displaystyle B\subseteq X}$ bounded by elements of ${\displaystyle Y}$; that is, ${\displaystyle \mathcal{B}}$ is the set of all subsets ${\displaystyle B\subseteq X}$ such that for every ${\displaystyle y\in Y,}$

${\displaystyle |y{|}_B=\underset{x\in B}{sup}|⟨x,y⟩|<\mathrm{\infty }.}$

Then the strong topology ${\displaystyle \beta (Y,X,⟨\cdot ,\cdot ⟩)}$ on ${\displaystyle Y,}$ also denoted by ${\displaystyle b(Y,X,⟨\cdot ,\cdot ⟩)}$ or simply ${\displaystyle \beta (Y,X)}$ or ${\displaystyle b(Y,X)}$ if the pairing ${\displaystyle ⟨\cdot ,\cdot ⟩}$ is understood, is defined as the locally convex topology on ${\displaystyle Y}$ generated by the seminorms of the form

${\displaystyle |y{|}_B=\underset{x\in B}{sup}|⟨x,y⟩|,y\in Y,B\in \mathcal{B}.}$

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

${\displaystyle |f{|}_B=\underset{x\in B}{sup}|f(x)|,f\in X^{\prime },}$

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

Examples

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

See also

• Dual topology
• Dual system
• Reflexive space
• Polar topology – Dual space topology of uniform convergence on some sub-collection of bounded subsets
• Semi-reflexive space
• Strong dual space – Continuous dual space endowed with the topology of uniform convergence on bounded sets
• Topologies on spaces of linear maps

References

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

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