Dual pair

A dual pair $(E,F)$ consists of a real vector space $E$ and a vector space $F$ of linear functionals that separates the points of $E$. Its weak topology $\sigma (E,F)$ is the coarsest topology making every $f\in F$ continuous, and its continuous dual is exactly $F$.

For a separated dual pair, the continuous linear functionals for its weak topology are exactly evaluations by members of the paired space. Continuity bounds a functional on a neighbourhood defined by finitely many evaluations; scalar rescaling makes it vanish on their common kernel. It then factors through their finite-dimensional evaluation map and is a finite linear combination of them.

Let $Z\subseteq X^{\ast }$ separate points of a Banach space $X$. If the closed unit ball of $X$ is compact for $\sigma (X,Z)$ and $Z$ has the norm inherited from $X^{\ast }$, then the evaluation map $X\to Z^{\ast }$ is an isometric isomorphism. Goldstine theorem makes its unit-ball image weak-star dense, while the assumed compactness makes that image closed.

For $A\subseteq E$ in a dual pair $(E,F)$, define $A^{\circ }=\{f\in F:f(a)\le 1\text{ for all }a\in A\}$. Then