# three simple question about Banach space

If $X$ is a Banach space, and its closed unit ball is separable, then do we know that $X$ is separable?

If $X$ is a separable Banach space, then do we know that its closed unit ball is separable?

If $X$ is a Banach space, and its unit sphere is separable, then do we know that $X$ is separable too?

A topological space is separable if it has a countably dense subset.

This isn't obvious to me.

• For the second one, for any set $S$ that is countable and dense in $X$, $S\cap B$ is countable and dense in the closed unit ball $B$. – Clement C. Apr 3 '15 at 14:05
• (note that it is also true for the open unit ball, although it is not as straightforward.) – Clement C. Apr 3 '15 at 14:09
• @Cement C.I know that any open subset of a separable space is separable, but for the closed subset? – David Chan Apr 3 '15 at 14:10