Strong dual space

Let ${\displaystyle (X,Y,\langle \cdot ,\cdot \rangle )}$ 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

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

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 ${\displaystyle |y|_{B}<\infty }$ for all ${\displaystyle y\in C.}$ So a subset ${\displaystyle B\subseteq X}$ is called bounded if and only if

$${\displaystyle \sup _{x\in B}|\langle x,y\rangle |<\infty \quad {\text{ for all }}y\in Y.}$$

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 ${\displaystyle y\in Y,}$

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

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

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

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

$${\displaystyle |f|_{B}=\sup _{x\in B}|f(x)|,\qquad {\text{ where }}f\in X^{\prime },}$$

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 ${\displaystyle X_{\beta }^{\prime }.}$

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 ${\displaystyle X^{\prime }}$ is given by the polars:

$${\displaystyle B^{\circ }:=\left\{x^{\prime }\in X^{\prime }:\sup _{x\in B}\left|x^{\prime }(x)\right|\leq 1\right\}}$$

as ${\displaystyle B}$ ranges over ${\displaystyle {\mathcal {B}}}$). This is a locally convex topology that is given by the set of seminorms on ${\displaystyle X^{\prime }}$: ${\displaystyle \left|x^{\prime }\right|_{B}:=\sup _{x\in B}\left|x^{\prime }(x)\right|}$ as ${\displaystyle B}$ ranges over ${\displaystyle {\mathcal {B}}.}$

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