Vol. 27, No. 2, 1968

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A note on Eberlein’s theorem

Carl Louis DeVito

Vol. 27 (1968), No. 2, 261–263

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 Eis 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.

Mathematical Subject Classification
Primary: 46.01
Received: 12 October 1967
Published: 1 November 1968
Carl Louis DeVito