Normed Vector Space is Hausdorff Locally Convex Space

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

Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.

Then $\struct {X, \norm {\, \cdot \,} }$ is a Hausdorff locally convex space.

Proof
From Normed Vector Space is Locally Convex Space, $\struct {X, \norm {\, \cdot \,} }$ can be viewed as the locally convex space $\struct {X, \norm {\, \cdot \,} }$.

Now for $x \ne \mathbf 0_X$, we have:


 * $\norm x \ne 0$

from.

So $\struct {X, \norm {\, \cdot \,} }$ is a Hausdorff locally convex space.