Normed Vector Space is Hausdorff Locally Convex Space

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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 Norm Axiom $\text N 1$: Positive Definiteness.

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

$\blacksquare$