Definition:Open Set/Normed Vector Space

Definition

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

Let $U \subseteq X$.

Then $U$ is an open set in $V$ if and only if:

$\forall x \in U: \exists \epsilon \in \R_{>0}: \map {B_\epsilon} x \subseteq U$

where $\map {B_\epsilon} x$ is the open $\epsilon$-ball of $x$.

Sources

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