Characterization of Continuity of Linear Functional in Weak-* Topology

Theorem
Let $\GF \in \set {\R, \C}$.

Let $X$ be a topological vector space over $\GF$ with weak topology $w$.

Let $X^\ast$ be the topological dual space of $X$.

Let $w^\ast$ be the weak-$\ast$ topology on $X^\ast$.

Then $\phi : \struct {X^\ast, w^\ast} \to \GF$ is continuous there exists $x \in X$ such that:


 * $\phi = x^\wedge$

where $x^\wedge$ is the evaluation linear transformation evaluated at $x$.

That is:


 * $\struct {X^\ast, w^\ast}^\ast = \iota X$

Proof
This is precisely Continuity of Linear Functionals in Initial Topology on Vector Space Generated by Linear Functionals, taking $F = \set {x^\wedge : x \in X}$.