Today’s problem appeared as Problem 6 on the Texas A&M August 2021 Analysis Qual:
Problem 6. Show that if 
is a separable Banach space, then there exists a bounded injective linear map from into 
Solution: Note that with every Banach space we have the canonical embedding into the double dual 
Moreover, since is separable, the unit ball in 
is weak
metrizable, and by Banach-Alaouglu it is weak compact. Therefore, the unit ball in is weak separable, i.e. there exists a countable subset 
of functionals dense in the unit ball in the weak topology. More precisely, this implies that if 
then there exist a subsequence 
such that 
for all 
(this follows from the fact that sequences define the weak topology, since it is metrizable on bounded sets). Finally, let 
for all 
be a sequence such that 
I claim that 
given by 
is the desired map. It is clearly linear, and since 
Finally, to show injectivity, by linearity it suffices to show that the kernel is trivial. Indeed, suppose that 
for all 
for some 
Then, letting 
we note that 
for all 
But by the weak density of in the unit ball of 
we conclude that 
for all 
By the Hahn-Banach theorem (which implies the existence of nonzero linear functionals) and the injectivity of the canonical embedding into the double dual, this implies that 
so 
is an injective linear bounded map from into 
as desired.