It is proved that a Banach space X is a subspace of a weakly compactly generated Banach space if and only if, for every ε > 0, X can be covered by a countable collection of bounded closed convex symmetric sets where the weak~* closure in X~(**) of each of them lies within the distance ε from X. A new short functional-analytic proof of the known result that a continuous image of an Eberlein compact is Eberlein is given as a corollary.
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