Hostname: page-component-77c89778f8-cnmwb Total loading time: 0 Render date: 2024-07-21T09:26:49.140Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak Compactness and Vector Measures

Published online by Cambridge University Press:  20 November 2018

R. G. Bartle ,
N. Dunford  and
J. Schwartz
R. G. Bartle
Affiliation:
Yale University
N. Dunford
Affiliation:
Yale University
J. Schwartz
Affiliation:
Yale University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Introduction. It is the purpose of this paper to develop a Lebesgue theory of integration of scalar functions with respect to a countably additive measure whose values lie in a Banach space. The class of integrable functions reduces to the ordinary space of Lebesgue integrable functions if the measure is scalar valued. Convergence theorems of the Vitali and Lebesgue type are valid in the general situation. The desirability of such a theory is indicated by recent developments in spectral theory.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 7 , 1955 , pp. 289 - 305
Copyright
Copyright © Canadian Mathematical Society 1955

References

1. Alexiewicz, A., On sequences of operations (I), Studia Math., 11 (1949), 1–30.Google Scholar
2. Bartle, R. G., On compactness in functional analysis, Trans. Amer. Math. Soc, 79 (1955).Google Scholar
3. Dunford, N. and Pettis, B. J., Linear transformations on summable functions, Trans. Amer. Math. Soc, 47 (1940), 323–392.Google Scholar
4. Dunford, N. and Schwartz, J., Spectral theory; forthcoming book.Google Scholar
5. Eberlein, W. F., Weak compactness in Banach spaces (I), Proc. Nat. Acad. Sci. U.S.A., 33 (1947), 51–53.Google Scholar
6. Gelfand, I., Abstrakte Funktionen und lineare Operatoren, Mat. Sbornik (4)46 (1938), 235–284.Google Scholar
7. Grothendieck, A., Sur les applications linéares faiblement compactes d'espace du type C(K), Can. J. Math., 5 (1953), 129–173.Google Scholar
8. Halmos, P. R., Measure theory (New York, 1950).Google Scholar
8a. Hille, E., Functional analysis and semi-groups (Amer. Math. Soc. Colloquium Publications, vol. 31, 1948).Google Scholar
9. Kakutani, S., Concrete representation of abstract (L)-spaces and the mean ergodic theorem, Ann. Math. (2) 42 (1941), 523–537.Google Scholar
10. Kakutani, S., Concrete representation of abstract (M)-spaces, Ann. Math. (2) 42 (1941), 994–1024.Google Scholar
11. B. J. Pettis, , On integration in vector spaces, Trans. Amer. Math. Soc, 44 (1938), 277–304.Google Scholar
12. B. J. Pettis, , Absolutely continuous functions in vector spaces (Abstract), Bull. Amer. Math. Soc, 45 (1939), 677.Google Scholar
13. Phillips, R. S., On linear transformations, Trans. Amer. Math. Soc, 48 (1940), 516–541.Google Scholar
14. Radon, J., Über lineare Funktionaltransformationen und Funktionalgleichungen, Sitzber. Akad. Wiss. Wien, 128 (1919), 1083–1121.Google Scholar
15. Saks, S., On some functionals, Trans. Amer. Math. Soc, 35 (1933), 549–556, 965–970.Google Scholar
16. Sirvint, G., Weak compactness in Banach spaces, Studia Math., 11 (1949), 71–94.Google Scholar