The following is an easy Corollary of the equivalence of weak compactness and weak sequential compactness in the weak topology on a normed space $X$:
A subset $E$ of a normed space $X$ is weakly compact if and only if $E\cap Y$ is weakly compact for every norm-closed and separable subspace $Y$ of $X$.
Is it possible to prove this without using the equivalence of weak compactness and weak sequential compactness as can be obtained by the Eberlein-Smulian Theorem? And if so, I would be pleased if you would share your ideas or references.
Thanks in advance.