Definition:Everywhere Dense/Normed Vector Space

Definition

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

Let $Y \subseteq X$ be a subset of $X$.

Suppose:

$\forall x \in X: \forall \epsilon \in \R_{>0}: \exists y \in Y: \norm {x - y} < \epsilon$

Then $Y$ is (everywhere) dense in $X$.

Sources

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