Pearson distribution

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 $(X,Y,⟨,⟩)$ 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:

$\mathrm{\forall }y\in Y{\sup }_{x\in B}|⟨x,y⟩|<\mathrm{\infty }.$

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

$\left|\right|y|{|}_B={\sup }_{x\in B}|⟨x,y⟩|,y\in Y,B\in \mathcal{ℬ}.$

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

$\left|\right|f|{|}_B={\sup }_{x\in B}|f(x)|,f\in X^{\prime },$

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

Examples

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

References

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