Normed Vector Space is Closed in Itself
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Theorem
Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space.
Then $X$ is closed in $M$.
Proof
From Empty Set is Open in Normed Vector Space, $\O$ is open in $M$.
But:
- $X = \relcomp X \O$
where $\complement_X$ denotes the set complement relative to $X$.
The result follows by definition of closed set.
$\blacksquare$
Also see
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.3$: Normed and Banach spaces. Topology of normed spaces