Weak Topology on Topological Vector Space over Hausdorff Topological Field is Hausdorff

Theorem

Let $K$ be a Hausdorff topological field.

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

Then $\struct {X, w}$ is Hausdorff.

Proof

From the definition of the weak topology, if $x, y \in X$ have $x \ne y$, then $x - y \ne \mathbf 0_X$:

there exists $f \in X^\ast$ such that $\map f {x - y} \ne 0$.

Since $f$ is linear, we then have $\map f x \ne \map f y$.

Since $K$ is Hausdorff, we therefore obtain that $\struct {X, w}$ is Hausdorff from Initial Topology with respect to Point-Separating Family of Mappings onto Hausdorff Spaces is Hausdorff.

$\blacksquare$