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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.

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