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

Theorem
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 topological vector space.

Proof
Follows from combining:


 * Weak Topology on Topological Vector Space over Hausdorff Topological Field is Hausdorff
 * Vector Addition is Continuous in Weak Topology
 * Scalar Multiplication is Continuous in Weak Topology