no. 3
On certain barrelled normed spaces
Annales de l'Institut Fourier, Volume 29 (1979) no. 3, pp. 39-56

Let 𝒜 be a σ-algebra on a set X. If A belongs to 𝒜 let e(A) be the characteristic function of A. Let ℓ 0 ∞ (X,𝒜 be the linear space generated by {e(A):A∈𝒜} endowed with the topology of the uniform convergence. It is proved in this paper that if (E n ) is an increasing sequence of subspaces of ℓ 0 ∞ (X,𝒜) covering it, there is a positive integer p such that E p is a dense barrelled subspace of ℓ 0 ∞ (X,𝒜), and some new results in measure theory are deduced from this fact.

Soit 𝒜 une σ-algèbre dans un ensemble X. Si A appartient à 𝒜, soit e(A) la fonction caractéristique de A. Soit ℓ 0 ∞ (X,𝒜 l’espace vectoriel engendré par {e(A):A∈𝒜} avec la topologie de la convergence uniforme. On montre que si (E n ) est une suite croissante des sous-espaces de ℓ 0 ∞ (X,𝒜) dont l’union est ℓ 0 ∞ (X,𝒜) il existe un entier positif p, telle que E p est un sous-espace dense et tonnelé de ℓ 0 ∞ (X,𝒜). Quelques nouveaux résultats dans la théorie de la mesure sont déduits de ce fait.

Valdivia, Manuel. On certain barrelled normed spaces. Annales de l'Institut Fourier, Volume 29 (1979) no. 3, pp. 39-56. doi: 10.5802/aif.752
@article{AIF_1979__29_3_39_0,
     author = {Valdivia, Manuel},
     title = {On certain barrelled normed spaces},
     journal = {Annales de l'Institut Fourier},
     pages = {39--56},
     year = {1979},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {29},
     number = {3},
     doi = {10.5802/aif.752},
     zbl = {0379.46004},
     mrnumber = {81d:46006},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.752/}
}
TY  - JOUR
AU  - Valdivia, Manuel
TI  - On certain barrelled normed spaces
JO  - Annales de l'Institut Fourier
PY  - 1979
SP  - 39
EP  - 56
VL  - 29
IS  - 3
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.752/
DO  - 10.5802/aif.752
LA  - en
ID  - AIF_1979__29_3_39_0
ER  - 
%0 Journal Article
%A Valdivia, Manuel
%T On certain barrelled normed spaces
%J Annales de l'Institut Fourier
%D 1979
%P 39-56
%V 29
%N 3
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.752/
%R 10.5802/aif.752
%G en
%F AIF_1979__29_3_39_0

[1] G. Bennett and N.J. Kalton, Addendum to “FK-spaces containing c0”, Duke Math. J., 39, (1972), 819-821. | Zbl | MR

[2] R.B. Darst, On a theorem of Nikodym with applications to weak convergence and von Neumann algebra, Pacific Jour. of Math., V. 23, No 3, (1967), 473-477. | Zbl | MR

[3] A. Grothendieck, Espaces vectoriels topologiques, Departamento de Matemática da Universidade de Sao Paulo, Brasil, 1954. | Zbl | MR

[4] I. Labuda, Exhaustive measures in arbitrary topological vector spaces, Studia Math., LVIII, (1976), 241-248. | Zbl | MR

[5] M. Valdivia, Sobre el teorema de la gráfica cerrada, Collectanea Math., XXII, Fasc. 1, (1971), 51-72. | Zbl

[6] M. Valdivia, On weak compactness, Studia Math., XLIX, (1973), 35-40. | Zbl | MR

Cited by Sources: