Definition:Closed Set/Normed Vector Space

Definition

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

Let $F \subset X$.

Definition 1

$F$ is closed in $V$ if and only if its complement $X \setminus F$ is open in $V$.

Definition 2

$F$ 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: $\S 1.3$: Normed and Banach spaces. Topology of normed spaces