Separable Metric Space and Separable Real Normed Space

In the setting of Separable Real Normed Spaces and Hahn–Banach: Notation, let $(X,d)$ be a metric space.

(i) Separable metric space. $(X,d)$ is separable if there is a countable subset $D\subseteq X$ that is dense in $(X,d)$.

(ii) Separable normed space. A real normed space $V$ is separable if the metric space $(V,d_{V})$, with $d_{V}$ its norm metric, is separable.