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.

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     author = {Valdivia, Manuel},
     title = {On certain barrelled normed spaces},
     journal = {Annales de l'Institut Fourier},
     pages = {39--56},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {29},
     number = {3},
     year = {1979},
     doi = {10.5802/aif.752},
     zbl = {0379.46004},
     mrnumber = {81d:46006},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.752/}
}
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.752/

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

[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. | MR | Zbl

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

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

[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. | MR | Zbl

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