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Summation and uncountable bi-orthogonal systems in locally convex spaces

Published online by Cambridge University Press:  20 January 2009

Harry F. Joiner II
Harry F. Joiner II
Affiliation:
University of Massachusetts, Amherst, Massachusetts 01002, U.S.A.
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The purpose of this paper is to extend to locally convex spaces and to uncountable systems several well-known results concerning infinite series, biorthogonal sequences, and Schauder bases. Section 2 gives three extensions of the theorem of Orlicz (10) and Pettis (11) and some lemmas that will be needed later. The third section introduces the notions of a summability basis and a summability basis of subspaces, and two main theorems are proved, including a simplification of Retherford and McArthur's proof (12) of a theorem of Nikol'skiĭ (9). Section 4 investigates the positive cone of an uncountable biorthogonal system, particularly conditions equivalent to the regularity of this cone.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 19 , Issue 1 , March 1974 , pp. 1 - 10
Copyright
Copyright © Edinburgh Mathematical Society 1974

References

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