If X is separable, then ball $X^*$ is weak-star metrizable.
135387
Ball $X^*$ is weak-star metrizable iff X is separable in Conway's book.
31002
Proving that the unit ball in $\ell^2(\mathbb{N})$ is non-compact
98756
Unit ball in $C[0,1]$ not sequentially compact
148001
In Hilbert space: $x_n → x$ if and only if $x_n \to x$ weakly and $\Vert x_n \Vert → \Vert x \Vert$.
16892
Reference request: product Borel $\sigma$-algebra of non-separable metric spaces
17733
$X$ second countable locally compact Hausdorff implies $C(X)$ separable?
47031
Isometric to Dual implies Hilbertable?
119649
Every metrizable space with a countable dense subset has a countable basis
132095
Motivation for the term "separable" in topology
32819
Looking for examples of first countable, compact spaces which is not separable
6491
Not every open set in $\mathbb{R}^n$, for $n\geq2$, is a disjoint union of countable open balls.
12467
Why does this proof work: Closed unit ball in $C_0$ is not compact
3708
How can I prove that compact subset iff sequentially compact in second countable topological spaces?
93419
Let $\mathcal A$ be a collection of pairwise disjoint subsets of a $\sigma$-finite measure space,Show that $\mathcal A$ is at most countable.
19856
Prove that a subset of a separable set is itself separable
105791
Does separability imply the Lindelöf property?
95074
Separability of $l^{p}$ spaces