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The space of Pettis integrable functions is barrelled


Authors: Lech Drewnowski, Miguel Florencio and Pedro J. Paúl
Journal: Proc. Amer. Math. Soc. 114 (1992), 687-694
MSC: Primary 46E40; Secondary 46A08, 46G10
DOI: https://doi.org/10.1090/S0002-9939-1992-1107271-2
MathSciNet review: 1107271
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Abstract: It is well known that the normed space of Pettis integrable functions from a finite measure space to a Banach space is not complete in general. Here we prove that this space is always barrelled; this tells us that we may apply two important results to this space, namely, the Banach-Steinhaus uniform boundedness principle and the closed graph theorem. The proof is based on a theorem stating that a quasi-barrelled space having a convenient Boolean algebra of projections is barrelled. We also use this theorem to give similar results for the spaces of Bochner integrable functions.


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  • [1] P. Antosik and J. Burzyk, Sequential conditions for barrelledness and bornology, Bull. Acad. Polon. Sci. Sér. Sci. Math. 35 (1987), 457-459. MR 939007 (89d:46001)
  • [2] J. Diestel and J. J. Uhl, Vector measures, Math Surveys and Monographs, vol. 15, Amer. Math. Soc., Providence, RI, 1977. MR 0453964 (56:12216)
  • [3] L. Drewnowski, M. Florencio, and P. J. Paúl, Uniform boundedness of operators and barrelledness in spaces with Boolean algebras of projections. IV Convegno di Analisi Reale e Teoria de Misura, Capri, Italy, 1990, (to appear).
  • [4] E. Hille and R. S. Phillips, Functional analysis and semi-groups, Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, RI, 1957. MR 0089373 (19:664d)
  • [5] L. Janicka and N. J. Kalton, Vector measures of infinite variation, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), 239-241. MR 0444889 (56:3235)
  • [6] G. Köthe, Topological vector spaces, I, II, Springer-Verlag, Berlin, Heidelberg, and New York, 1969, 1979. MR 0248498 (40:1750)
  • [7] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, II, Springer-Verlag, Berlin, Heidelberg, and New York, 1979. MR 540367 (81c:46001)
  • [8] J. A. López Molina, The dual and bidual of an echelon Köthe space, Collect. Math. 31 (1980), 159-191. MR 611558 (83f:46028)
  • [9] G. G. Lorentz and D. G. Wertheim, Representation of linear functionals on Köthe spaces, Canad. J. Math 5 (1953), 568-575. MR 0058119 (15:324h)
  • [10] P. Pérez Carreras and J. Bonet, Barrelled locally convex spaces, North-Holland Mathematics Studies, vol. 131 North-Holland, Amsterdam, New York, Oxford, and Tokyo, 1987. MR 880207 (88j:46003)
  • [11] B. J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), 277-304. MR 1501970
  • [12] K. Reiher, Weigthed inductive and projective limits of normed Köthe function spaces, Resultate Math. 13 (1988), 147-161. MR 928147 (89d:46034)
  • [13] W. J. Ricker and H. H. Schaefer, The uniformly closed algebra generated by a complete Boolean algebra of projections, Math. Z. 201 (1989), 429-439. MR 999738 (91b:47081)
  • [14] G. E. F. Thomas, Totally summable functions with values in locally convex spaces, Measure Theory (Proc. Conf. Oberwolfach, 1975), Lecture Notes in Math., vol. 541, Springer-Verlag, Berlin, Heidelberg, and New York, 1976, pp. 117-131. MR 0450505 (56:8799)
  • [15] R. Welland, On Köthe spaces, Trans. Amer. Math. Soc. 112 (1964), 267-277. MR 0172110 (30:2336)
  • [16] A. C. Zaanen, Integration, North-Holland, Amsterdam, 1967. MR 0222234 (36:5286)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1107271-2
Keywords: Barrelled spaces, Pettis integral, Bochner integral, Boolean algebras of projections
Article copyright: © Copyright 1992 American Mathematical Society

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