A note on Eberlein’s theorem

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 tr 9 weak* continuity on E r 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.