Locally Convex Space is Topological Vector Space

Theorem

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

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

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

Corollary

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 Vector Addition on Locally Convex Space is Continuous, vector addition on $X$ is continuous.

From Scalar Multiplication on Locally Convex Space is Continuous, scalar multiplication on $X$ is continuous.

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

$\blacksquare$