Subset of separable Hilbert space

Question

I have a separable Hilbert space H. How can one prove that the closed unit ball B is separable?

I'm told it's trivial but I can't see it.

My initial idea is to take the set A (dense set of H) and intersect it with B. This new set is clearly constable and it's closure is in B. but how can I show that B is in th closure of this set?

Thanks for any help

Answer 1

Note that metric spaces are regular.

Show that if $B$ is the closure of an open subset of a separable regular space $H$ and that $A$ is a countable dense subset of $H$ then $A\cap B$ is a countable dense subset of $B$ in the topology inherited from $H$.

Consider two cases:

1. $P$ is a point in the interior $B\,\backslash\, \beta\{B\}$ of $B$.
2. $P$ is a point on the boundary $\beta\{B\}\subseteq H$

Case 1: Let $U$ be an open set in $B$ containing $P$ but no point of the boundary of $B$ in $H$. Then since $U$ is also an open set in $H$ it contains a point of $A$.

Case 2: Let $U$ be an open set in $H$ containing $P$. Then $U\cap(B\,\backslash\, \beta\{B\})$ is an open set in $H$ as well as an open set in $B$ so it contains a point of $A\cap B\subset A$.

Thus $B$ is separable.