Characterization of Continuity of Linear Functional in Weak Topology
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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 $F = X^\ast$.
$\blacksquare$