A note on a theorem of J. Diestel and B. Faires
Authors:
J. C. Ferrando and M. López Pellicer
Journal:
Proc. Amer. Math. Soc. 115 (1992), 10771081
MSC:
Primary 46G10; Secondary 46A08
MathSciNet review:
1091179
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Abstract: Applying a property concerning certain coverings of that always contain some elements that are barrelled and dense in , we generalize a localization theorem of M. Valdivia, relative to vector bounded finitely additive measures (Theorem 1), and obtain two different generalizations of a theorem of J. Diestel and B. Faires ensuring that certain finitely additive measures are countably additive (Theorems 2 and 3). The original proof of the quoted theorem of Diestel and Faires uses a theorem of Rosenthal that is not required in our proof of Theorem 3. This avoids imposing over the Valdivia's spaces defining the measure range space, the condition that they do not contain a copy of .
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 L. Drewnowski, An extension of a theorem of Rosenthal on operators acting from , Studia Math. 57 (1976), 209215. MR 0423116 (54:11097)
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 J. C. Ferrando, and M. LópezPellicer, Strong barrelledness properties in and bounded finite additive measures. Math. Ann. 287 (1990), 727736. MR 1066827 (91i:46002)
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 M. Levin, and S. A. Saxon, A note in the inheritance of properties of locally convex spaces by subspaces of countable codimension, Proc. Amer. Math. Soc. 29 (1971), 97102. MR 0280973 (43:6692)
 [5]
 P. Pérez Carreras, and J. Bonet Barrelled locally convex spaces, North Holland Math. Studies, vol. 131, Amsterdam, New York, Oxford, 1987. MR 880207 (88j:46003)
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 M. Valdivia, On certain barrelled normed spaces. Ann. Inst. Fourier (Grenoble) 29 (1979), 3956. MR 552959 (81d:46006)
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 , Mackey convergence and the closed graph theorem Arch. Math. 25 (1974), 649656. MR 0374856 (51:11052)
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 , Sobre el teorema de la gráfica cerrada. Collect. Math. 22 (1971), 5172.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199210911795
PII:
S 00029939(1992)10911795
Keywords:
Closed graph theorems,
dual locally complete spaces,
and spaces,
barrelled spaces,
finitely (countably) additive vector measure,
bounded vector measure
Article copyright:
© Copyright 1992
American Mathematical Society
