Definition:Separable Space/Normed Vector Space
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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