Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 106 / 4 / a

The useful forms of the Hahn-Banach separation theorem in a real locally convex space are:

The Banach-Alaoglu theorem says that the closed unit ball of $E^{\ast }$ is compact in $\sigma (E^{\ast },E)$. Goldstine theorem says that the canonical image of the closed unit ball of a normed space $E$ is weak-star dense in the closed unit ball of $E^{\ast \ast }$.

Give $Z$ its norm inherited from $X^{\ast }$ and defineIt is linear and contractive, and it is injective because $Z$ separates points. Goldstine's theorem followed by restriction from $X^{\ast }$ to $Z$ shows that $J(B_X)$ is weak-star dense in $B_{Z^{\ast }}$: a functional on $Z$ first extends norm-preservingly to $X^{\ast }$, and elements of $B_X$ approximate that extension on every finite subset of $Z$.

The topology induced by $\sigma (Z^{\ast },Z)$ on $J(B_X)$ is exactly the given topology $\sigma (X,Z)$. By hypothesis $B_X$ is compact, so $J(B_X)$ is weak-star compact and therefore closed in the Hausdorff space $B_{Z^{\ast }}$. Density now givesThus $J$ is surjective and maps closed unit ball onto closed unit ball, so it is an isometric isomorphism. Hence $X$ is a dual space. This is the compact norming dual-pair criterion.