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