Strong dual space

Let ${\displaystyle (X,Y,\left\langle \cdot ,\cdot \right\rangle )}$ be a dual system of vector spaces over the field ${\displaystyle {\mathbb {F} }}$ of real (${\displaystyle {\mathbb {R} }}$) or complex (${\displaystyle {\mathbb {C} }}$) numbers. Note that neither X nor Y has a topology so we define a subset B of X to be bounded if and only if ${\displaystyle \sup _{x\in B}\left|\left\langle x,y\right\rangle \right|<\infty }$ for all ${\displaystyle y\in Y}$. This is equivalent to the usual notion of bounded subsets when X is given the weak topology induced by Y, which is a Hausdorff locally convex topology. The definition of the strong dual topology now proceeds as in the case of a TVS.

Note that if X is a TVS whose continuous dual space separates point on X, then X is part of a canonical dual system ${\displaystyle \left(X,X^{\prime },\left\langle \cdot ,\cdot \right\rangle \right)}$ where ${\displaystyle \left\langle x,x^{\prime }\right\rangle$ :=x^{\prime }(x)} .

Suppose that X is a topological vector space (TVS) over the field ${\displaystyle {\mathbb {F} }}$ of real (${\displaystyle {\mathbb {R} }}$) or complex (${\displaystyle {\mathbb {C} }}$) numbers. Let ${\displaystyle {\mathcal {B}}}$ be any fundamental system of bounded sets of X (i.e. a set of bounded subsets of X such that every bounded subset of X is a subset of some ${\displaystyle B\in {\mathcal {B}}}$); the set of all bounded subsets of X trivially forms a fundamental system of bounded sets of X. A basis of closed neighborhoods of the origin in ${\displaystyle X^{\prime }}$ is given by the polars:

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

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

If X is normable then so is ${\displaystyle X_{b}^{\prime }}$ and ${\displaystyle X_{b}^{\prime }}$ will in fact be a Banach space. If 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.