Vol. 22, No. 2, 1967

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On a theorem of Orlicz and Pettis

Charles Wisson McArthur

Vol. 22 (1967), No. 2, 297–302

In this paper a direct proof of the following theorem of Orlicz, Pettis, and Grothendieck is given.

Theorem 1. In a locally convex Hausdorff space each subseries of a series converges with respect to the initial topology of the space if and only if each subseries of the series converges with respect to the weak topology of the space.

In a second theorem each of three additional conditions is shown to be equivalent to subseries convergence in complete locally convex Hausdorff spaces. Two of these equivalence are known for Banach spaces. The third condition, a weak compactness condition on the unordered partial sums of the series, is new even for Banach spaces. It is a consequence of the first theorem that a weak unconditional basis for a weak sequentially complete locally convex Hausdorff space is an unconditional basis.

Mathematical Subject Classification
Primary: 46.01
Received: 17 June 1965
Published: 1 August 1967
Charles Wisson McArthur