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Let $X$ be a Banach space and $S\subseteq X$ be a subspace such that the unit sphere of $S$ is weakly compact. If $Y^*=X$ for some separable Banach space $Y,$ is it true that $S$ is separable?

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    $\begingroup$ Exercise: If $X$ is an infinite dimensional normed space, then its unit sphere is weakly dense in its unit ball. Hint: First prove that zero is in the weak closure of the unit sphere. $\endgroup$ – Bill Johnson Jun 18 '19 at 13:09
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    $\begingroup$ Maybe the OP wants to consider the unit ball rather than the unit sphere, which in light of Bill's comments is more reasonable to consider? $\endgroup$ – Dirk Werner Jun 18 '19 at 19:38
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    $\begingroup$ Now, every $Y$ as above is a quotient of $L_1[0,1]$, and hence every $X$ as above is a subspace of $L_\infty$. Thus, $S$ is a weakly compact subset of $L_\infty$ and therefore norm separable. $\endgroup$ – Dirk Werner Jun 18 '19 at 19:44
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    $\begingroup$ Correction: Not $S$ is weakly compact, but its unit ball... $\endgroup$ – Dirk Werner Jun 18 '19 at 20:39
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Assume that Samya Kumar Ray meant that the unit ball of $S$ is weakly compact. If $S$ is really a sphere, the fact that the answer is "Yes" follows from the argument outlined by Bill Johnson (and the additional assumptions are not needed).

The answer is "yes" anyway. Possibly a bit more straightforward argument than the one of Dirk Werner: the weak$^*$ topology on the unit ball of $S$ is weaker than the weak topology, so by the well-known result on compacts this implies that on the unit ball of $S$ weak$^*$ topology is the same as the weak topology. It is well-known and easy to show that bounded sets in the dual to a separable Banach space are weak$^*$ separable. Thus the unit ball of $S$ is weak separable. It remains to prove the easy statement that Banach spaces whose balls are weakly separable are separable.

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I found a proof using a theorem in "Topcis In Banach Space Theory" by Albiac and Kalton. Theorem: Let $X$ be a Banach space. The Dual $X^*$ contains a sequence $(x_n^*)$ having the property $X_n^*(x)=0$ for all $n\geq 1$ implies $x=0.$ Then any weakly compact subset of $X$ is metrizable in weak topology.

Now to the original question take a sequence $(y_n)\subseteq Y\subseteq Y^{**}=X^*$ such that $(y_n)$ is dense in $Y$. Clearly $(y_n)$ is total. The result follows applying the above theorem.

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