Weak Cauchy sequences in L ∞ ( μ , X )

Georg Schlüchtermann

Studia Mathematica (1995)

  • Volume: 116, Issue: 3, page 271-281
  • ISSN: 0039-3223

Abstract

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For a finite and positive measure space Ω,∑,μ characterizations of weak Cauchy sequences in L ∞ ( 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 ∞ ( μ , X ) is discussed.

How to cite

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Schlüchtermann, Georg. "Weak Cauchy sequences in $L_{∞}(μ,X)$." Studia Mathematica 116.3 (1995): 271-281. <http://eudml.org/doc/216233>.

@article{Schlüchtermann1995,
abstract = {For a finite and positive measure space Ω,∑,μ characterizations of weak Cauchy sequences in $L_\{∞\}(μ,X)$, 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 $L_\{∞\}(μ,X)$ is discussed.},
author = {Schlüchtermann, Georg},
journal = {Studia Mathematica},
keywords = {essentially bounded vector-valued functions; measure space; weak Cauchy sequences; distinction between Asplund and conditionally weakly compact subsets},
language = {eng},
number = {3},
pages = {271-281},
title = {Weak Cauchy sequences in $L_\{∞\}(μ,X)$},
url = {http://eudml.org/doc/216233},
volume = {116},
year = {1995},
}

TY - JOUR
AU - Schlüchtermann, Georg
TI - Weak Cauchy sequences in $L_{∞}(μ,X)$
JO - Studia Mathematica
PY - 1995
VL - 116
IS - 3
SP - 271
EP - 281
AB - For a finite and positive measure space Ω,∑,μ characterizations of weak Cauchy sequences in $L_{∞}(μ,X)$, 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 $L_{∞}(μ,X)$ is discussed.
LA - eng
KW - essentially bounded vector-valued functions; measure space; weak Cauchy sequences; distinction between Asplund and conditionally weakly compact subsets
UR - http://eudml.org/doc/216233
ER -

References

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  1. [AU] K. T. Andrews and J. J. Uhl, On weak compactness in L ∞ ( μ , X ) , Indiana Univ. Math. J. 30 (1981), 907-915. Zbl0473.46022
  2. [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. Zbl0491.46010
  3. [Bo] R. Bourgin, Geometric Aspects of Convex Sets with the Radon-Nikodým Property, Lecture Notes in Math. 993, Springer, 1983. Zbl0512.46017
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  6. [DRS] J. Diestel, W. Ruess and W. Schachermayer, On weak compactness in L 1 ( μ , X ) , Proc. Amer. Math. Soc. 118 (1993), 447-453. Zbl0785.46037
  7. [DU] J. Diestel and J. Uhl, Vector Measures, Math. Surveys 15, Amer. Math. Soc., 1977. 
  8. [Ku] C. Kuratowski, Topologie I, Monograf. Mat., Warszawa, 1933. 
  9. [La] H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer, Berlin, 1974. Zbl0285.46024
  10. [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. Zbl0547.46009
  11. [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. 
  12. [S1] G. Schlüchtermann, Renorming in the space of Bochner integrable functions L 1 ( μ , X ) , Manuscripta Math. 73 (1991), 397-409. 
  13. [S2] G. Schlüchtermann, On weakly compact operators, Math. Ann. 292 (1992), 263-266. Zbl0735.47012
  14. [S3] G. Schlüchtermann, Weak compactness in L ∞ ( μ , X ) , J. Funct. Anal. 125, (1994), 379-388. Zbl0828.46036
  15. [Ta] M. Talagrand, Weak Cauchy sequences in L 1 ( E ) , Amer. J. Math. 106 (1984), 703-724. Zbl0579.46025

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