Hausdorff LCTVS: A°° (in E) is the Closed Absolutely Convex Hull of A

If E is a Hausdorff LCTVS then A°° (in E) is the Closed Absolutely Convex Hull of A

Proposition 1: Let $E$ be a Hausdorff locally convex topological vector space so that $(E, E')$, $(E', E)$ are dual pairs and let $A \subseteq E$. Then $A^{\circ \circ} = \overline{\mathrm{abs \: conv}(A)}$, that is, the polar of $A^{\circ \circ}$ (in $E$) is the closed absolutely convex hull of $A$.

• Proof: Since $E$ is a Hausdorff locally convex topological vector space, both $(E, E')$ and $(E', E)$ are dual pairs and are such that $E \subseteq E \subseteq E'^*$.

• So by the theorem on the If E ⊆ G ⊆ F* and (E, F) is a Dual Pair then A°° is the σ(G, F)-Closed Absolutely Convex Hull of A page:

(1)

\begin{align} \quad A^{\circ \circ} = \overline{\mathrm{abs \: conv}(A)}^{\sigma(E, E')} \end{align}

• But since $\mathrm{abs \: conv}(A)$ is absolutely convex, it is convex and since $\sigma(E, E')$ is a topology of the dual pair $(E, E')$, we have that the $\sigma(E, E')$ closure of $\mathrm{abs \: conv}(A)$ is the same as the closure of $\mathrm{abs \: conv}(A)$, and so:

(2)

\begin{align} \quad A^{\circ \circ} = \overline{\mathrm{abs \: conv}(A)} \quad \blacksquare \end{align}