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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.

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    $\begingroup$ 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 $B$. $\endgroup$ – Clement C. Apr 3 '15 at 14:05
  • $\begingroup$ (note that it is also true for the open unit ball, although it is not as straightforward.) $\endgroup$ – Clement C. Apr 3 '15 at 14:09
  • $\begingroup$ @Cement C.I know that any open subset of a separable space is separable, but for the closed subset? $\endgroup$ – David Chan Apr 3 '15 at 14:10
  • $\begingroup$ See the comment above: take your favorite dense and countable subset S$ in $X$, and show it is dense in the closed unit ball as well. $\endgroup$ – Clement C. Apr 3 '15 at 14:13
  • $\begingroup$ A metric space is separable if and only if it is second countable. Second countability passes to subspaces.The second is trival. $\endgroup$ – David Chan Apr 3 '15 at 14:13
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The answer is yes in the 3 cases.

  • For the first one, consider the subset

$$\bigcup_{n\geq 1} nE$$

Where E is a countable dense subset of the unit ball.

Now you can show it's dense in all of X (basically, if you have a point x, you just have to show it can be approached by points of $(\lfloor \|x\| \rfloor +1)E$ )

  • For the second one, just consider $E\cap \bar{B}(0,1)$ and you can show it's dense in $\bar{B}(0,1)$

  • For the last one, consider the subset

$$\bigcup_{q \in \mathbb{Q}} qE$$

Where E is a countable dense subset of the unit sphere.

Now you can show it's dense in all of X. Take a sequence of $\mathbb{Q}$ that converge to $\|x\|$ and a sequence of the spere that converge to $\frac{x}{\|x\|}$ and it's easy to construct a sequence that converge to x

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