The space is not -complete

Manuel Valdivia

Annales de l'institut Fourier (1977)

  • Volume: 27, Issue: 4, page 29-43
  • ISSN: 0373-0956

Abstract

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Certain classes of locally convex space having non complete separated quotients are studied and consequently results about -completeness are obtained. In particular the space of L. Schwartz is not -complete where denotes a non-empty open set of the euclidean space .

How to cite

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Valdivia, Manuel. "The space $D(U)$ is not $B_r$-complete." Annales de l'institut Fourier 27.4 (1977): 29-43. <http://eudml.org/doc/74340>.

@article{Valdivia1977,
abstract = {Certain classes of locally convex space having non complete separated quotients are studied and consequently results about $B_r$-completeness are obtained. In particular the space of L. Schwartz $\{\bf D\}(\Omega )$ is not $B_r$-complete where $\Omega $ denotes a non-empty open set of the euclidean space $R^m$.},
author = {Valdivia, Manuel},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {4},
pages = {29-43},
publisher = {Association des Annales de l'Institut Fourier},
title = {The space $D(U)$ is not $B_r$-complete},
url = {http://eudml.org/doc/74340},
volume = {27},
year = {1977},
}

TY - JOUR
AU - Valdivia, Manuel
TI - The space $D(U)$ is not $B_r$-complete
JO - Annales de l'institut Fourier
PY - 1977
PB - Association des Annales de l'Institut Fourier
VL - 27
IS - 4
SP - 29
EP - 43
AB - Certain classes of locally convex space having non complete separated quotients are studied and consequently results about $B_r$-completeness are obtained. In particular the space of L. Schwartz ${\bf D}(\Omega )$ is not $B_r$-complete where $\Omega $ denotes a non-empty open set of the euclidean space $R^m$.
LA - eng
UR - http://eudml.org/doc/74340
ER -

References

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  1. [1] A. GROTHENDIECK, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc., No. 16 (1966). Zbl0123.30301
  2. [2] G. KHÖTE, Topological Vector Spaces I. Berlin-Heidelberg-New York, Springer 1969. Zbl0179.17001
  3. [3] A. PIETSCH, Nuclear locally convex spaces. Berlin-Heidelberg-New York, Springer 1972. Zbl0308.47024MR50 #2853
  4. [4] V. PTAK, Completeness and open mapping theorem, Bull. Soc. Math. France, 86 (1958), 41-74. Zbl0082.32502MR21 #4345
  5. [5] D. A. RAIKOV, On B-complete topological vector groups, Studia Math., 31 (1968), 295-306. Zbl0183.40202
  6. [6] O. G. SMOLJANOV, The space D is not hereditarily complete, Izv. Akad. Nauk SSSR, Ser. Math., 35 (3) (1971), 686-696; Math. USSR Izvestija, 5 (3) (1971), 696-710. Zbl0249.46020
  7. [7] M. VALDIVIA, On countable locally convex direct sums, Arch. d. Math., XXVI, 4 (1975), 407-413. Zbl0312.46014MR52 #1241
  8. [8] M. VALDIVIA, On Br-completeness, Ann. Inst. Fourier, Grenoble 25, 2 (1975), 235-248. Zbl0301.46004MR53 #3634
  9. [9] M. VALDIVIA, Mackey convergence and the closed graph theorem, Arch. d. Math., XXV, 6 (1974), 649-656. Zbl0297.46006MR51 #11052
  10. [10] M. VALDIVIA, The space of distributions D′(Ω) is not Br-complete, Math. Ann., 211 (1974), 145-149. Zbl0288.46033MR51 #6406

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