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Remarks on Weak Compactness in L1(μ,X)

Published online by Cambridge University Press:  18 May 2009

J. Diestel
J. Diestel
Affiliation:
Kent State UniversityKentOhio 44242, U.S.A.
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Let (Ω,Σ,μ) be a finite measure space and X a Banach space. Denote by L1 (μ,X) the Banach space of (equivalence classes of) μ-strongly measurable X-valued Bochner integrable functions f:Ω→X normed by

The problem of characterizing the relatively weakly compact subsets of L1(Ω, X) remains open. It is known that for a bounded subset of L1(μ, X) to be relatively weakly compact it is necessary that the set be uniformly integrable; recall that K ⊆ L1, (μ, X) is uniformly integrable whenever given ε >0 there exists δ > 0 such that if μ (E) ≦ δ then ∫E∥f∥ dμ ≦ δ, for all f ∈ K. S. Chatterji has noted that in case X is reflexive this condition is also sufficient [4]. At present unless one assumes that both X and X* have the Radon-Nikodym Property (see [1]), a rather severe restriction which, for purposes of potential applicability, is tantamount to assuming reflexivity, no good sufficient conditions for weak compactness in L1(μ, X) exist. This note puts forth such sufficient conditions; the basic tool is the recent factorization method of W. J. Davis, T. Figiel, W. B. Johnson and A. Pelczynski [3].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 18 , Issue 1 , January 1977 , pp. 87 - 91
Copyright
Copyright © Glasgow Mathematical Journal Trust 1977

References

REFERENCES

1.Batt, J., On weak compactness in spaces of vector-valued measures and Bochner integrable functions in connection with the Radon-Nikodym property of Banach spaces, Rev, Roumaine Math. Pures Appl. 19 (1974), 285–304.Google Scholar
2.Brooks, J. K., Equicontinuous sets of measures and applications to Vitali's integral convergence theorem and control measures, Advances in Math. 10 (1973), 165–171.CrossRefGoogle Scholar
3.Davis, W. J., Figiel, T., Johnson, W. B. and Pelczynski, A., Factoring weakly compact operators, J. Functional Analysis 17 (1974), 311–327.CrossRefGoogle Scholar
4.Diestel, J. and Uhl, J. J., Theory of vector measures, to appear.Google Scholar
5.Dinculeanu, N., Vector measures, VEB Deutscher Verlag der Wissenschaften (Berlin, 1966).Google Scholar
6.Gretsky, N. and Uhl, J. J. Jr, Bounded linear operators on Banach function spaces of vectorvalued functions, Trans. Amer. Math. Soc. 167 (1972), 263–277.CrossRefGoogle Scholar
7.Grothendieck, A., Sur les applications linéaires faiblement compactes d'espaces du type C(K), Canad. J. Math. 5 (1953), 129–173.CrossRefGoogle Scholar
8.Grothendieck, A., Produits tensoriels topologiques et espaces nucleaires, Mem. Amer. Math. Soc, No. 16 (1955).Google Scholar
9.Przeworska-Rolewicz, D. and Rolewicz, S., Equations in linear spaces, Monografje Matemateczyne (Polish Scientific Publishers, 1968).Google Scholar