Topological Vector Space over Topological Field remains Topological Vector Space with Weak Topology/Corollary

Corollary

Let $K$ be a Hausdorff topological field.

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

Then $\struct {X, w}$ is a Hausdorff topological vector space.

Proof

From Topological Vector Space over Topological Field remains Topological Vector Space with Weak Topology, $\struct {X, w}$ is a topological vector space.

From Weak Topology on Topological Vector Space over Hausdorff Topological Field is Hausdorff, $\struct {X, w}$ is a Hausdorff topological vector space.

$\blacksquare$