$X$ is a normed linear space such that for some compact $K\subseteq X$ , $\operatorname{span} K$ is dense in $X$ then is $X$ separable?

Question

Let $X$ be a normed linear space which is separable. Then I know that there exists a compact subset $K$ of $X$ such that $\operatorname{span} K$ is dense in $X$ (in fact we can also find compact and countable subset $K$). My question is, is the converse true? i.e. if $X$ is a normed linear space such that there exist a compact $K\subseteq X$ such that $\operatorname{span} K$ is dense in $X$ then is $X$ separable? Please help. Thanks in advance.

Answer 1

Yes. Since $K$ is a compact metric space it is separable; now if $C$ is a countable dense subset of $K$ the closed span of $C$ is the same as the closed span of $K$.