On B r -completeness

Manuel Valdivia

Annales de l'institut Fourier (1975)

  • Volume: 25, Issue: 2, page 235-248
  • ISSN: 0373-0956

Abstract

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In this paper it is proved that if { E n } n = 1 ∞ and { F n } n = 1 ∞ are two sequences of infinite-dimensional Banach spaces then H = ⊕ n = 1 ∞ E n × ∏ n = 1 ∞ F n is not B r -complete. If { E n } n = 1 ∞ and { F n } n = 1 ∞ 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.

How to cite

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Valdivia, Manuel. "On $B_r$-completeness." Annales de l'institut Fourier 25.2 (1975): 235-248. <http://eudml.org/doc/74226>.

@article{Valdivia1975,
abstract = {In this paper it is proved that if $\lbrace E_n\rbrace ^\infty _\{n=1\}$ and $\lbrace F_n\rbrace ^\infty _\{n=1\}$ are two sequences of infinite-dimensional Banach spaces then $H = \big ( \oplus ^\infty _\{n=1\} E_n\big ) \times \prod ^\infty _\{n=1\}F_n$ is not $B_r$-complete. If $\lbrace E_n\rbrace ^\infty _\{n=1\}$ and $\lbrace F_n\rbrace ^\infty _\{n=1\}$ are also reflexive spaces there is on $H$ a separated locally convex topology $\{\cal F\}$, coarser than the initial one, such that $H[\{\cal F\}]$ 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.},
author = {Valdivia, Manuel},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {2},
pages = {235-248},
publisher = {Association des Annales de l'Institut Fourier},
title = {On $B_r$-completeness},
url = {http://eudml.org/doc/74226},
volume = {25},
year = {1975},
}

TY - JOUR
AU - Valdivia, Manuel
TI - On $B_r$-completeness
JO - Annales de l'institut Fourier
PY - 1975
PB - Association des Annales de l'Institut Fourier
VL - 25
IS - 2
SP - 235
EP - 248
AB - In this paper it is proved that if $\lbrace E_n\rbrace ^\infty _{n=1}$ and $\lbrace F_n\rbrace ^\infty _{n=1}$ are two sequences of infinite-dimensional Banach spaces then $H = \big ( \oplus ^\infty _{n=1} E_n\big ) \times \prod ^\infty _{n=1}F_n$ is not $B_r$-complete. If $\lbrace E_n\rbrace ^\infty _{n=1}$ and $\lbrace F_n\rbrace ^\infty _{n=1}$ are also reflexive spaces there is on $H$ a separated locally convex topology ${\cal F}$, coarser than the initial one, such that $H[{\cal F}]$ 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.
LA - eng
UR - http://eudml.org/doc/74226
ER -

References

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  2. [2] M. De WILDE, Réseaux dans les espaces linéaires à semi-normes, Mém. Soc. Royale des Sc. de Liège, 5e, série, XVIII, 2, (1969). Zbl0199.18103
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  4. [4] J. DIEUDONNE, Sur les propriétés de permanence de certains espaces vectoriels topologiques, Ann. Soc. Polon. Math., 25 (1952), 50-55. Zbl0049.08202
  5. [5] G. KÖTHE, Topological Vector Spaces I, Berlin-Heidelberg-New York, Springer (1969). Zbl0179.17001
  6. [6] G. KÖTHE, Über die Vollstandigkeit einer Klasse lokalconveser Räume, Math. Nachr. Z., 52, (1950), 627-630. Zbl0036.07901
  7. [7] J.T. MARTI, Introduction to the theory of Bases, Berlin-Heidelberg-New York, Springer (1969). Zbl0191.41301
  8. [8] V. PTAK, Completeness and the open mapping theorem, Bull. Soc. Math. France., 86 (1958), 41-47. Zbl0082.32502
  9. [9] M. VALDIVIA, A class of bornological barrelled spaces which are not ultrabornological, Math. Ann., 194 (1971), 43-51. Zbl0207.42701
  10. [10] M. VALDIVIA, Some examples on quasi-barrelled spaces, Ann. Inst. Fourier, 22 (1972), 21-26. Zbl0226.46005
  11. [11] M. VALDIVIA, On nonbornological barrelled spaces, Ann. Inst. Fourier, 22 (1972), 27-30. Zbl0226.46006
  12. [12] M. VALDIVIA, On countable strict inductive limits, Manuscripta Mat., 11 (1971), 339-343. Zbl0278.46002
  13. [13] M. VALDIVIA, A hereditary property in locally convex spaces, Ann. Inst. Fourier, 21 (1971), 1-2. Zbl0205.40903
  14. [14] M. VALDIVIA, Absolutely convex sets in barrelled spaces, Ann. Inst. Fourier, 21 (1971), 3-13. Zbl0205.40904
  15. [15] M. VALDIVIA, The space of distributions Dʹ (Ω) is not Br-complete, Math. Ann., 211 (1974), 145-149. Zbl0288.46033

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