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 dense in $X$.

Let $Y$ be a countable set:

$Y = \set {y_1, y_2, \ldots}$

Suppose:

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


Then $X$ is separable.


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