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I came across in my textbook the term: norm separable. I looked in the textbook and online and could not find a definition.

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The meaning is that the topology generated by the norm is a separable topology. Namely there exists a countable set $\{v_n\mid n\in\mathbb N\}$ such that for every $v$ and every $\varepsilon>0$ there exists $n$ such that $\|v-v_n\|<\varepsilon$.

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It means that the space is separable in the metric topology generated by the norm: it has a countable dense subset in that topology.

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