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