Normed Vector Space is Closed in Itself

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

• Space is Closed in Itself

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.3$: Normed and Banach spaces. Topology of normed spaces