Locally Convex Space is Topological Vector Space/Corollary

Theorem

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

Let $\struct {X, \PP}$ be a Hausdorff locally convex space over $\GF$ equipped with the standard topology $\tau$.

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

Proof

From Locally Convex Space is Topological Vector Space, $\struct {X, \tau}$ is a topological vector space.

From Locally Convex Space is Hausdorff iff induces Hausdorff Topology, $\struct {X, \tau}$ is a Hausdorff space.

So $\struct {X, \tau}$ is a Hausdorff topological vector space.

$\blacksquare$