On $B_r$-completeness

Valdivia, Manuel

doi:10.5802/aif.564

B_{r}

-completeness

Valdivia, Manuel

Annales de l'Institut Fourier, Tome 25 (1975) no. 2, pp. 235-248.

Résumé
Abstract

Dans le présent article il est démontré que si ${E_{n}}_{n = 1}^{\infty}$ et ${F_{n}}_{n = 1}^{\infty}$ sont deux suites d’espaces de Banach de dimension infinie, alors $H = (\oplus_{n = 1}^{\infty} E_{n}) \times \prod_{n = 1}^{\infty} F_{n}$ n’est pas $B_{r}$ -complet. Il est démontré aussi que si ${E_{n}}_{n = 1}^{\infty}$ et ${F_{n}}_{n = 1}^{\infty}$ sont de plus des espaces réflexifs il y a sur $H$ une topologie localement convexe et séparée $ℱ$ moins fine que l’initiale, telle que $H [ℱ]$ est un espace tonnelé et bornologique, qui n’est pas limite inductive d’espaces de Baire. On donne aussi d’autres résultats sur la $B_{r}$ -complétude et les espaces bornologiques.

In this paper it is proved that if ${E_{n}}_{n = 1}^{\infty}$ and ${F_{n}}_{n = 1}^{\infty}$ are two sequences of infinite-dimensional Banach spaces then $H = (\oplus_{n = 1}^{\infty} E_{n}) \times \prod_{n = 1}^{\infty} F_{n}$ is not $B_{r}$ -complete. If ${E_{n}}_{n = 1}^{\infty}$ and ${F_{n}}_{n = 1}^{\infty}$ are also reflexive spaces there is on $H$ a separated locally convex topology $ℱ$ , coarser than the initial one, such that $H [ℱ]$ is a bornological barrelled space which is not an inductive limit of Baire spaces. It is given also another results on $B_{r}$ -completeness and bornological spaces.

MR Zbl | 1 citation dans Numdam

DOI : 10.5802/aif.564

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Valdivia, Manuel. On $B_r$-completeness. Annales de l'Institut Fourier, Tome 25 (1975) no. 2, pp. 235-248. doi : 10.5802/aif.564. https://www.numdam.org/articles/10.5802/aif.564/

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