Definition:Separable Space/Normed Vector Space

Definition

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

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

Let $Y$ be countable set and (everywhere) dense in $X$.

In other words, suppose $Y = \set {y_i : i \in \N}$ such that:

$\forall x \in X : \forall \epsilon \in \R_{> 0} : \epsilon > 0 : \exists y_{n \mathop \in \N} \in Y : \norm {y_n - x} < \epsilon$

Then $X$ is separable.

Sources

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