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Left Cauchy Integral Bases in Linear Topological Spaces

Published online by Cambridge University Press:  20 November 2018

James A. Dyer
James A. Dyer*
Affiliation:
Iowa State University, Ames, Iowa
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The purpose of this paper is to consider a representation for the elements of a linear topological space in the form of a σ-integral over a linearly ordered subset of V; this ordered subset is what will be called an L basis. The formal definition of an L basis is essentially an abstraction from ideas used, often tacitly, in proofs of many of the theorems concerning integral representations for continuous linear functionals on function spaces.

The L basis constructed in this paper differs in several basic ways from the integral basis considered by Edwards in [5]. Since the integrals used here are of Hellinger type rather than Radon type one has in the approximating sums for the integral an immediate and natural analogue to the partial sum operators of summation basis theory.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 4 , December 1970 , pp. 431 - 439
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Arsove, M. B., The Payley-Wiener theorem in metric linear spaces, Pacifi. J. Math. 10 (1960), 365-379.Google Scholar
2. Arsove, M. B. and Edwards, R. E., Generalized bases in topological linear spaces, Studia Math. 19(1960), 95-113.Google Scholar
3. Bessaga, C. and Pelczynski, A., Wlasnosci baz wprzestrzeniach typu B0 , Prace Mat. 3 (1959), 123-142.Google Scholar
4. Dieudonné, J., On bi-orthogonal systems, Michigan Math. J. 2 (1954), 7-20.Google Scholar
5. Edwards, R. E., Integral bases in inductive limit spaces, Pacifi. J. Math. 10 (1960), 797-812.Google Scholar
6. Hildebrandt, T. H., Introduction to the theory of integration, Academic Press, New York, 1963.Google Scholar
7. Kaltenborn, H. S., Linear functional operations on functions having discontinuities of the first kind, Bull. Amer. Math. Soc. 40 (1934), 702-708.Google Scholar
8. McArthur, C. W., The weak basis theorem, Colloq. Math. 17 (1967), 71-76.Google Scholar
9. Retherford, J. R. and McArthur, C. W., Some remarks on bases in linear topological spaces, Math. Ann. 164 (1966), 38-41.Google Scholar
10. Retherford, J. R., Bases, basic sequences and reflexivity of linear topological spaces, Math. Ann. 164 (1966), 280-285.Google Scholar
11. Singer, I., Basic sequences and reflexivity of Banach spaces, Studia Math. 21 (1962), 351-369.Google Scholar
12. Wilansky, A., Functional analysis, Blaisdell, Waltham, Mass., 1964.Google Scholar