User:Caliburn/s/fa/Characterization of Separable Normed Vector Space

Theorem
Let $\mathbb F \in \set {\R, \C}$.

Let $\struct {X, \norm \cdot}$ be a normed vector space over $\Bbb F$.


 * $(1): \quad$ $X$ is separable
 * $(2): \quad$ $S_X = \set {x \in X : \norm x = 1}$ is separable
 * $(3): \quad$ there exists a countable set $\set {x_n : n \in \N} \subseteq X$ such that the closed linear span of $\set {x_n : n \in \N}$ is $X$.