Weak Cauchy sequences in $L_{∞}(μ,X)$

Georg Schlüchtermann

ArticleOriginal scientific text

Title

Weak Cauchy sequences in $L_{\infty} (μ, X)$

Authors ¹

Affiliations

Mathematisches Institut der Ludwig-Maximilians-Universität, Theresienstrasse 39, 80333 München, Germany

Abstract

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.

[AU] K. T. Andrews and J. J. Uhl, On weak compactness in $L_{\infty} (μ, X)$ , Indiana Univ. Math. J. 30 (1981), 907-915.
[BH] J. Batt and W. Hiermeyer, On compactness in $L_{p} (μ, X)$ in the weak topology and in the topology $σ (L_{p} (μ, X), L_{q} (μ, X \cdot))$ , Math. Z. 182 (1983), 409-432.
[Bo] R. Bourgin, Geometric Aspects of Convex Sets with the Radon-Nikodým Property, Lecture Notes in Math. 993, Springer, 1983.
[Di] J. Diestel, Sequences and Series in Banach Spaces, Springer, Berlin, 1984.
[Din] N. Dinculeanu, Vector Measures, Deutscher Verlag Wiss., Berlin, 1966.
[DRS] J. Diestel, W. Ruess and W. Schachermayer, On weak compactness in $L_{1} (μ, X)$ , Proc. Amer. Math. Soc. 118 (1993), 447-453.
[DU] J. Diestel and J. Uhl, Vector Measures, Math. Surveys 15, Amer. Math. Soc., 1977.
[Ku] C. Kuratowski, Topologie I, Monograf. Mat., Warszawa, 1933.
[La] H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer, Berlin, 1974.
[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.
[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.
[S1] G. Schlüchtermann, Renorming in the space of Bochner integrable functions $L_{1} (μ, X)$ , Manuscripta Math. 73 (1991), 397-409.
[S2] G. Schlüchtermann, On weakly compact operators, Math. Ann. 292 (1992), 263-266.
[S3] G. Schlüchtermann, Weak compactness in $L_{\infty} (μ, X)$ , J. Funct. Anal. 125, (1994), 379-388.
[Ta] M. Talagrand, Weak Cauchy sequences in $L^{1} (E)$ , Amer. J. Math. 106 (1984), 703-724.

Title

Weak Cauchy sequences in μL∞(μ,X)

Affiliations

Abstract

Bibliography

Weak Cauchy sequences in $L_{\infty} (μ, X)$