Definition:Closed Set/Normed Vector Space/Definition 2

Definition

Let $V = \struct{X, \norm{\,\cdot\,} }$ be a normed vector space.

Let $F \subset X$.

$F$ is closed (in $V$) if and only if every limit point of $F$ is also a point of $F$.

That is: if and only if $F$ contains all its limit points.

Also see

• Equivalence of Definitions of Closed Set in Normed Vector Space

• Results about closed sets can be found here.

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 1.4$: Normed and Banach spaces. Sequences in a normed space; Banach spaces