The space of Pettis integrable functions is barrelled

Drewnowski, Lech; Florencio, Miguel; Paúl, Pedro J.

doi:10.1090/S0002-9939-1992-1107271-2

The space of Pettis integrable functions is barrelled
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by Lech Drewnowski, Miguel Florencio and Pedro J. Paúl PDF

Proc. Amer. Math. Soc. 114 (1992), 687-694 Request permission

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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