Locally Convex Space is Topological Vector Space/Corollary
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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.
$\blacksquare$