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

Theorem

Let $K$ be a topological field.

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

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

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

Follows from combining:

Vector Addition is Continuous in Weak Topology

Scalar Multiplication is Continuous in Weak Topology

$\blacksquare$