Weak Cauchy sequences in $L_{∞}(μ,X)$

Schlüchtermann, Georg

Weak Cauchy sequences in

L_{\infty} (μ, X)

Schlüchtermann, Georg

Studia Mathematica, Tome 113 (1995), p. 271-281 / Harvested from The Polish Digital Mathematics Library

Access to full text
Full (PDF)

Résumé

For a finite and positive measure space Ω,∑,μ characterizations of weak Cauchy sequences in $L_{\infty} (μ, X)$ , the space of μ-essentially bounded vector-valued functions f:Ω → X, are presented. The fine distinction between Asplund and conditionally weakly compact subsets of $L_{\infty} (μ, X)$ is discussed.

Publié le : 1995-01-01

Zbl 0851.46024

EUDML-ID : urn:eudml:doc:216233

@article{bwmeta1.element.bwnjournal-article-smv116i3p271bwm,
     author = {Georg Schl\"uchtermann},
     title = {Weak Cauchy sequences in $L\_{$\infty$}($\mu$,X)$
            },
     journal = {Studia Mathematica},
     volume = {113},
     year = {1995},
     pages = {271-281},
     zbl = {0851.46024},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv116i3p271bwm}
}

Schlüchtermann, Georg. Weak Cauchy sequences in $L_{∞}(μ,X)$
            . Studia Mathematica, Tome 113 (1995) pp. 271-281. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv116i3p271bwm/

Bibliographie

[00000] [AU] K. T. Andrews and J. J. Uhl, On weak compactness in $L_{\infty} (μ, X)$ , Indiana Univ. Math. J. 30 (1981), 907-915. | Zbl 0473.46022

[00001] [BH] J. Batt and W. Hiermeyer, On compactness in $L_{p} (μ, X)$ in the weak topology and in the topology $σ (L_{p} (μ, X), L_{q} (μ, X *))$ , Math. Z. 182 (1983), 409-432. | Zbl 0491.46010

[00002] [Bo] R. Bourgin, Geometric Aspects of Convex Sets with the Radon-Nikodým Property, Lecture Notes in Math. 993, Springer, 1983. | Zbl 0512.46017

[00003] [Di] J. Diestel, Sequences and Series in Banach Spaces, Springer, Berlin, 1984.

[00004] [Din] N. Dinculeanu, Vector Measures, Deutscher Verlag Wiss., Berlin, 1966.

[00005] [DRS] J. Diestel, W. Ruess and W. Schachermayer, On weak compactness in $L_{1} (μ, X)$ , Proc. Amer. Math. Soc. 118 (1993), 447-453. | Zbl 0785.46037

[00006] [DU] J. Diestel and J. Uhl, Vector Measures, Math. Surveys 15, Amer. Math. Soc., 1977.

[00007] [Ku] C. Kuratowski, Topologie I, Monograf. Mat., Warszawa, 1933.

[00008] [La] H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer, Berlin, 1974. | Zbl 0285.46024

[00009] [RSU] L. H. Riddle, E. Saab and J. J. Uhl, Sets with the weak Radon-Nikodým property in dual Banach spaces, Indiana Univ. Math. J. 32 (1983), 527-541. | Zbl 0547.46009

[00010] [RU] L. H. Riddle and J. J. Uhl, Martingales and the fine line between Asplund spaces and spaces not containing a copy of $ℓ_{1}$ , in: Martingale Theory in Harmonic Analysis and Banach Spaces, Lecture Notes in Math. 939, Springer, 1983, 145-156.

[00011] [S1] G. Schlüchtermann, Renorming in the space of Bochner integrable functions $L_{1} (μ, X)$ , Manuscripta Math. 73 (1991), 397-409.

[00012] [S2] G. Schlüchtermann, On weakly compact operators, Math. Ann. 292 (1992), 263-266. | Zbl 0735.47012

[00013] [S3] G. Schlüchtermann, Weak compactness in $L_{\infty} (μ, X)$ , J. Funct. Anal. 125, (1994), 379-388. | Zbl 0828.46036

[00014] [Ta] M. Talagrand, Weak Cauchy sequences in $L^{1} (E)$ , Amer. J. Math. 106 (1984), 703-724. | Zbl 0579.46025