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.
$\blacksquare$