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 $f : X \to \GF$ be a linear functional.

Then $f$ is $w$-continuous if and only if $f \in X^\ast$.

That is:

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

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Proof

This is precisely Continuity of Linear Functionals in Initial Topology on Vector Space Generated by Linear Functionals, taking $F = X^\ast$.

$\blacksquare$