A Banach space $X$ is relfexive iff $X^*$ is reflexive.
6955
A Banach space is reflexive if and only if its dual is reflexive
87655
Consequence of the Hahn-Banach Theorem
124458
Quotient of a Banach space $X$ gets quotient topology under standard norm induced from $X$.
148001
In Hilbert space: $x_n → x$ if and only if $x_n \to x$ weakly and $\Vert x_n \Vert → \Vert x \Vert$.
26112
Inclusion of $l^p$ space for sequences
98634
What is a counterexample for If $x\in\overline{A},$ $x$ is not necessarily a limit of a sequence in $A$?
22295
Subspace of a separable space is separable
19991
Weak convergence, together with convergence of norms, implies strong convergence in a Hilbert space.
63608
Cauchy sequence in a normed space
134859
Let $(X, \mathcal{T})$ be a metrizable space. Prove that $X$ Lindelöf implies that $X$ is separable.
20120
Converging series in Banach space
115350
An example of a Banach space isomorphic but not isometric to a dual Banach space
102879
$\operatorname{Ext}$ and injectives, respectively projectives
2056
A space is $T_{0}$ if and only if the closures of singletons are distinct
133266
If $(X, \mathcal{T})$ has a countable subbasis, then it has a countable basis
133504
Suppose $X$ is a metric space and $S \subseteq X.$ Then, $S^o=\{x \in X~|~dist(x,S^c)>0\}$.
47810
$C(X)$ is separable when $X$ is compact?