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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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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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