If $X^*$ is separable $\space \implies X$ is separable.In which conditions the opposite direction is true?
8334
A reflexive Banach space is separable iff its dual is separable
142145
strictly positive elements in $C^*$-algebra
95928
Independence and orthogonality
125623
Subspaces of separable normed spaces
134703
Let $X$ be a connected space, $f:X\to X$ continuous involution, $g:X\to\mathbb{R}$ continuous. Prove that $\exists$ $x\in X$ such that $g(x)=g(f(x))$.
135387
Ball $X^*$ is weak-star metrizable iff X is separable in Conway's book.
127166
Specific example of a space that is separable but not second countable.
121106
Parallelogram law valid in banach spaces?
57055
Are $|X|$ and $\operatorname{sgn}(X)$ independent?
130998
Is $\exists x \in A ~:~ P(x)$ the same as $\exists x ~:~ x \in A \implies P(x)$
49385
Proving $\forall x (A\to B) \to(A \to \forall x B):x\notin \mbox{free}(A)$ in a Hilbert system where it is not an axiom
132095
Motivation for the term "separable" in topology
153355
Given $S \in B(Y^{*}, X^{*})$, does there exist $T\in B(X,Y)$ such that $S=T^{*}$?
125776
If space of bounded operators L(V,W) is Banach, V nonzero, then W is Banach (note direction of implication)
115963
Need help understanding separable polynomials
22295
Subspace of a separable space is separable
127522
Space $\mathcal{L}^p(X, \Sigma, \mu)$ is separable iff $(\Sigma, \rho_\Delta)$ is separable