This paper is concerned
with locally convex spaces which are closed, separable subspaces of their strong
biduals. Let E be a space of this type. We first prove that, for an element of
E′′, weak∗ continuity on E′ is equivalent to sequential weak∗ continuity
on the convex, strongly bounded subsets of E′. We then prove Eberlein’s
theorem for spaces of this type; i.e., we prove that, for the weakly closed
subsets of E, countable weak compactness coincides with weak compactness.
Finally, we show that the separability hypothesis in our first theorem is
necessary.
