Definition:Closed Set/Normed Vector Space/Definition 1

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 its complement $X \setminus F$ is open in $V$.

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