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Exhaustive operators and vector measures

Published online by Cambridge University Press:  20 January 2009

N. J. Kalton
N. J. Kalton
Affiliation:
University College of Swansea, Singleton Park, Swansea SA2 8PP
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Let S be a compact Hausdorff space and let Φ: C(S)→E be a linear operator defined on the space of real-valued continuous functions on S and taking values in a (real) topological vector space E. Then Φ is called exhaustive (7) if given any sequence of functions fn ∈ C(S) such that fn ≧ 0 and

then Φ(fn)→0 If E is complete then it was shown in (7) that exhaustive maps are precisely those which possess regular integral extensions to the space of bounded Borel functions on S; this is equivalent to possessing a representation

where μ is a regular countably additive E-valued measure defined on the σ-algebra of Borel subsets of S.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 19 , Issue 3 , March 1975 , pp. 291 - 300
Copyright
Copyright © Edinburgh Mathematical Society 1975

References

REFERENCES

(1) Arsove, M. G. and Edwards, R. E., Generalised bases in topological linear spaces, Studia Math. 19 (1960), 95–113.CrossRefGoogle Scholar
(2) Bessaga, C. and PelczyŃski, A., On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151–164.CrossRefGoogle Scholar
(3) Diestel, J., Applications of bases and weak compactness to vector measures and vectorial integration, Rev. Roum. Math. Pures et Appl. 18 (1973), 211–224.Google Scholar
(4) Drewnowski, L., Another note on Kalton's theorem (to appear).Google Scholar
(5) Dunford, N. and Schwartz, J. T., Linear Operators, Vol. I (Interscience, New York, 1958).Google Scholar
(6) Grothendieck, A., Sur les applications lindaires faiblement compactes d'espaces du type C(K), Canad. J. Math. 5 (1953), 129–173.CrossRefGoogle Scholar
(7) Kalton, N. J., Topologies on Riesz groups and applications to measure theory, Proc. London Math. Soc. (3) 18 (1974), 253–273.CrossRefGoogle Scholar
(8) Kalton, N. J., Basic sequences in F-spaces and their applications, Proc. Edinburgh Math. Soc. 19 (1974), 151–167.CrossRefGoogle Scholar
(9) Kelley, J. L., General Topology (van Nostrand, New York, 1955).Google Scholar
(10) Köthe, G., Topological Vector Spaces (Springer, Berlin, 1969).Google Scholar
(11) Labuda, I., Sur quelques théoremès du type d'Orlicz-Pettis I, Bull. Acad. Polon. Sci. 21 (1973), 127–132.Google Scholar
(12) Mcarthur, C. W., On a theorem of Orlicz and Pettis, Pacific J. Math. 22 (1967), 297–302.CrossRefGoogle Scholar
(13) Orlicz, W., Absolute continuity of set functions with respect to a finitely sub- additive measure, Prace Mat. 14 (1970), 101–118.Google Scholar
(14) Pelczynski, A., Banach spaces on which every unconditionally converging operator is weakly compact, Bull. Acad. Polon. Sci. 10 (1962), 641–648.Google Scholar
(15) Robertson, A. P., On unconditional convergence in topological vector spaces, Proc. Roy. Soc. Edinburgh, Sect. A 68 (1969), 145–157.Google Scholar
(16) Rosenthal, H. P., On relatively disjoint families of measures, with some applications to Banach space theory. Studia Math. 37 (1971), 13–36.CrossRefGoogle Scholar
(16) Schwartz, L., Un théoremè de la convergence dans les espaces Lp, 0 ≦p<∞, C.R. Acad. Sci. Paris, Ser. A 268 (1969), 704–706.Google Scholar
(18) Thomas, E., Sur le théorème d'Orlicz et un problème de M. Laurent Schwartz, C.R. Acad. Sci. Paris, Ser. A. 267 (1968), 7–10.Google Scholar
(19) Whitley, R. J., Projecting m onto c0, Amer. Math. Monthly 73 (1966), 285–286.CrossRefGoogle Scholar