On nonbornological barrelled spaces
Annales de l'Institut Fourier, Tome 22 (1972) no. 2, p. 27-30
On démontre que si E est le produit topologique d’une famille non dénombrable d’espaces tonnelés de dimension non nulle, il existe un nombre infini de sous-espaces tonnelés de E, qui ne sont pas bornologiques. Un résultat semblable est obtenu si l’on change “tonnelé” en “infratonnelé”.
If E is the topological product of a non-countable family of barrelled spaces of non-nulle dimension, there exists an infinite number of non-bornological barrelled subspaces of E. The same result is obtained replacing “barrelled” by “quasi-barrelled”.
@article{AIF_1972__22_2_27_0,
     author = {Valdivia, Manuel},
     title = {On nonbornological barrelled spaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {22},
     number = {2},
     year = {1972},
     pages = {27-30},
     doi = {10.5802/aif.410},
     zbl = {0226.46006},
     mrnumber = {49 \#1050},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1972__22_2_27_0}
}
Valdivia, Manuel. On nonbornological barrelled spaces. Annales de l'Institut Fourier, Tome 22 (1972) no. 2, pp. 27-30. doi : 10.5802/aif.410. http://www.numdam.org/item/AIF_1972__22_2_27_0/

[1] N. Bourbaki, Sur certains spaces vectoriels topologiques, Ann. Inst. Fourier, 5-16 (1950). | Numdam | MR 13,137d | Zbl 0042.35302

[2] J. Dieudonné, Recent development in the theory of locally convex spaces, Bull. Amer. Math. Soc., 59, 495-512 (1953). | MR 15,963a | Zbl 0053.25701

[3] J. Dieudonné, Sur les propriétés de permanence de certains espaces vectoriels topologiques, Ann. Soc. Polon. Math., 25, 50-55 (1952). | MR 15,38b | Zbl 0049.08202

[4] Y. Komura, Some examples on linear topological spaces, Math. Ann., 153, 150-162 (1964). | MR 32 #2884 | Zbl 0149.33604

[5] L. Nachbin, Topological vector spaces of continuous functions, Proc. Nat. Acad. Sci., USA, 40, 471-474 (1954). | MR 16,156h | Zbl 0055.09803

[6] T. Shirota, On locally convex vector spaces of continuous functions, Proc. Jap. Acad., 30, 294-298 (1954). | MR 16,275d | Zbl 0057.33801