Strong dual space

Strong dual topology

Throughout, all vector spaces will be assumed to be over the field ${\displaystyle \mathbb{F}}$ of either the real numbers ${\displaystyle \mathbb{R}}$ or complex numbers ${\displaystyle \mathbb{C}.}$

Definition from a dual system

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

Neither ${\displaystyle X}$ nor ${\displaystyle Y}$ has a topology so say 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 ${\displaystyle y\in C.}$ So a subset ${\displaystyle B\subseteq X}$ is called bounded if and only if $${\displaystyle \underset{x\in B}{sup}|⟨x,y⟩|<\mathrm{\infty }\text{ for all }y\in Y.}$$ This is equivalent to the usual notion of bounded subsets when ${\displaystyle X}$ is given the weak topology induced by ${\displaystyle Y,}$ which is a Hausdorff locally convex topology.

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

The definition of the strong dual topology now proceeds as in the case of a TVS. Note that if ${\displaystyle X}$ is a TVS whose continuous dual space separates point on ${\displaystyle X,}$ then ${\displaystyle X}$ is part of a canonical dual system ${\displaystyle \left(X,X^{\mathrm{\prime }},⟨\cdot ,\cdot ⟩\right)}$ where ${\displaystyle ⟨x,x^{\mathrm{\prime }}⟩:=x^{\mathrm{\prime }}(x).}$ In the special case when ${\displaystyle X}$ is a locally convex space, the strong topology on the (continuous) dual space ${\displaystyle X^{\mathrm{\prime }}}$ (that is, on the space of all continuous linear functionals ${\displaystyle f:X\to \mathbb{F}}$) is defined as the strong topology ${\displaystyle \beta \left(X^{\mathrm{\prime }},X\right),}$ and it coincides with the topology of uniform convergence on bounded sets in ${\displaystyle X,}$ i.e. with the topology on ${\displaystyle X^{\mathrm{\prime }}}$ generated by the seminorms of the form $${\displaystyle |f{|}_B=\underset{x\in B}{sup}|f(x)|,\text{ where }f\in X^{\mathrm{\prime }},}$$ where ${\displaystyle B}$ runs over the family of all bounded sets in ${\displaystyle X.}$ The space ${\displaystyle X^{\mathrm{\prime }}}$ with this topology is called strong dual space of the space ${\displaystyle X}$ and is denoted by ${\displaystyle X_{\beta }^{\mathrm{\prime }}.}$

Definition on a TVS

Suppose that ${\displaystyle X}$ is a topological vector space (TVS) over the field ${\displaystyle \mathbb{F}.}$ Let ${\displaystyle \mathcal{B}}$ be any fundamental system of bounded sets of ${\displaystyle X}$; that is, ${\displaystyle \mathcal{B}}$ is a family of bounded subsets of ${\displaystyle X}$ such that every bounded subset of ${\displaystyle X}$ is a subset of some ${\displaystyle B\in \mathcal{B}}$; the set of all bounded subsets of ${\displaystyle X}$ forms a fundamental system of bounded sets of ${\displaystyle X.}$ A basis of closed neighborhoods of the origin in ${\displaystyle X^{\mathrm{\prime }}}$ is given by the polars: $${\displaystyle B^{\circ }:=\left\{x^{\mathrm{\prime }}\in X^{\mathrm{\prime }}:\underset{x\in B}{sup}\left|x^{\mathrm{\prime }}(x)\right|\le 1\right\}}$$ as ${\displaystyle B}$ ranges over ${\displaystyle \mathcal{B}}$). This is a locally convex topology that is given by the set of seminorms on ${\displaystyle X^{\mathrm{\prime }}}$: ${\displaystyle {\left|x^{\mathrm{\prime }}\right|}_B:=\underset{x\in B}{sup}\left|x^{\mathrm{\prime }}(x)\right|}$ as ${\displaystyle B}$ ranges over ${\displaystyle \mathcal{B}.}$

If ${\displaystyle X}$ is normable then so is ${\displaystyle X_b^{\mathrm{\prime }}}$ and ${\displaystyle X_b^{\mathrm{\prime }}}$ will in fact be a Banach space. If ${\displaystyle X}$ is a normed space with norm ${\displaystyle \Vert \cdot \Vert }$ then ${\displaystyle X^{\mathrm{\prime }}}$ has a canonical norm (the operator norm) given by ${\displaystyle \Vert x^{\mathrm{\prime }}\Vert :=\underset{\Vert x\Vert \le 1}{sup}\left|x^{\mathrm{\prime }}(x)\right|}$; the topology that this norm induces on ${\displaystyle X^{\mathrm{\prime }}}$ is identical to the strong dual topology.