# On certain barrelled normed spaces

Annales de l'institut Fourier (1979)

- Volume: 29, Issue: 3, page 39-56
- ISSN: 0373-0956

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topValdivia, Manuel. "On certain barrelled normed spaces." Annales de l'institut Fourier 29.3 (1979): 39-56. <http://eudml.org/doc/74425>.

@article{Valdivia1979,

abstract = {Let $\{\cal A\}$ be a $\sigma $-algebra on a set $X$. If $A$ belongs to $\{\cal A\}$ let $e(A)$ be the characteristic function of $A$. Let $\ell ^\infty _0(X,\{\cal A\}$ be the linear space generated by $\lbrace e(A):A \in \{\cal A\}\rbrace $ 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 $\ell ^\infty _0(X,\{\cal A\})$ covering it, there is a positive integer $p$ such that $E_p$ is a dense barrelled subspace of $\ell ^\infty _0(X,\{\cal A\})$, and some new results in measure theory are deduced from this fact.},

author = {Valdivia, Manuel},

journal = {Annales de l'institut Fourier},

language = {eng},

number = {3},

pages = {39-56},

publisher = {Association des Annales de l'Institut Fourier},

title = {On certain barrelled normed spaces},

url = {http://eudml.org/doc/74425},

volume = {29},

year = {1979},

}

TY - JOUR

AU - Valdivia, Manuel

TI - On certain barrelled normed spaces

JO - Annales de l'institut Fourier

PY - 1979

PB - Association des Annales de l'Institut Fourier

VL - 29

IS - 3

SP - 39

EP - 56

AB - Let ${\cal A}$ be a $\sigma $-algebra on a set $X$. If $A$ belongs to ${\cal A}$ let $e(A)$ be the characteristic function of $A$. Let $\ell ^\infty _0(X,{\cal A}$ be the linear space generated by $\lbrace e(A):A \in {\cal A}\rbrace $ 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 $\ell ^\infty _0(X,{\cal A})$ covering it, there is a positive integer $p$ such that $E_p$ is a dense barrelled subspace of $\ell ^\infty _0(X,{\cal A})$, and some new results in measure theory are deduced from this fact.

LA - eng

UR - http://eudml.org/doc/74425

ER -

## References

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- [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. Zbl0189.44901MR38 #6360
- [3] A. GROTHENDIECK, Espaces vectoriels topologiques, Departamento de Matemática da Universidade de Sao Paulo, Brasil, 1954. Zbl0058.33401MR17,1110a
- [4] I. LABUDA, Exhaustive measures in arbitrary topological vector spaces, Studia Math., LVIII, (1976), 241-248. Zbl0365.46037MR55 #8789
- [5] M. VALDIVIA, Sobre el teorema de la gráfica cerrada, Collectanea Math., XXII, Fasc. 1, (1971), 51-72. Zbl0223.46009
- [6] M. VALDIVIA, On weak compactness, Studia Math., XLIX, (1973), 35-40. Zbl0243.46003MR48 #11969

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