Initial Topology on Normed Vector Space is Weak Topology

Theorem

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

Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\GF$.

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

Let $w$ be the initial topology on $X$ with respect to $X^\ast$.

Then $w$ is the weak topology on $X$.

Proof

From Normed Dual Space Separates Points, if $x \ne y$ then there exists $f \in X^\ast$ such that $\map f x \ne \map f y$.

That is, if $x \ne \mathbf 0_X$, there exists $f \in X^\ast$ such that $\map f x \ne 0$.

$\blacksquare$