I am trying to solve the following problem but I don't really getting it where to start from, which way to think. Any help would be appreciated.
If $(X,\|.\|)$ is a separable Banach space,
a) Why is the unit sphere $S=\{x\in X ;\|x\|=1\}$ separable in the relative topology?
b) If $(x_n)_{n\in \mathbb N } \subset S$ is a dense sequence and $T:\ell^1(\mathbb N) \to X $ defined by $T((a_n)_{n\in\mathbb N})=\sum_{n\in \mathbb N }a_n x_n$ , why is $T$ bounded and surjective ?
Is is true that $X$ is topologically isomorphic to $\ell^1(\mathbb {N})/\mathrm{Ker}(T)$?
Thanks