The space D(U) is not B r -complete
Annales de l'Institut Fourier, Tome 27 (1977) no. 4, pp. 29-43.

On étudie quelques classes d’espaces localement convexes avec quotients séparés et non-complets et en conséquence on obtient des résultats de B r -complétude. En particulier, l’espace de L. Schwartz D(Ω) n’est pas B r -complet, où Ω représente un ensemble non-vide de l’espace euclidien R m .

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 D(Ω) is not B r -complete where Ω denotes a non-empty open set of the euclidean space R m .

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     title = {The space $D(U)$ is not $B_r$-complete},
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Valdivia, Manuel. The space $D(U)$ is not $B_r$-complete. Annales de l'Institut Fourier, Tome 27 (1977) no. 4, pp. 29-43. doi : 10.5802/aif.671. https://aif.centre-mersenne.org/articles/10.5802/aif.671/

[1] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc., No. 16 (1966). | Zbl

[2] G. Khöte, Topological Vector Spaces I. Berlin-Heidelberg-New York, Springer 1969. | Zbl

[3] A. Pietsch, Nuclear locally convex spaces. Berlin-Heidelberg-New York, Springer 1972. | MR | Zbl

[4] V. Ptak, Completeness and open mapping theorem, Bull. Soc. Math. France, 86 (1958), 41-74. | Numdam | MR | Zbl

[5] D. A. Raikov, On B-complete topological vector groups, Studia Math., 31 (1968), 295-306.

[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. | Zbl

[7] M. Valdivia, On countable locally convex direct sums, Arch. d. Math., XXVI, 4 (1975), 407-413. | MR | Zbl

[8] M. Valdivia, On Br-completeness, Ann. Inst. Fourier, Grenoble 25, 2 (1975), 235-248. | Numdam | MR | Zbl

[9] M. Valdivia, Mackey convergence and the closed graph theorem, Arch. d. Math., XXV, 6 (1974), 649-656. | MR | Zbl

[10] M. Valdivia, The space of distributions D′(Ω) is not Br-complete, Math. Ann., 211 (1974), 145-149. | EuDML | MR | Zbl

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