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On Banach spaces with Mazur's property

Published online by Cambridge University Press:  18 May 2009

Denny H. Leung
Denny H. Leung
Affiliation:
Department of Mathematics, The University of Texas at Austin, Austin, TX 78712
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A Banach space E is said to have Mazur's property if every weak* sequentially continuous functional in E” is weak* continuous, i.e. belongs to E. Such spaces were investigated in [5] and [9] where they were called d-complete and μB-spaces respectively. The class of Banach spaces with Mazur's property includes the WCG spaces and, more generally, the Banach spaces with weak* angelic dual balls [4, p. 564]. Also, it is easy to see that Mazur's property is inherited by closed subspaces. The main goal of this paper is to present generalizations of some results of [5] concerning the stability of Mazur's property with respect to forming some tensor products of Banach spaces. In particular, we show in Sections 2 and 3 that the spaces E ⊗εF and Lp(μ, E) inherit Mazur's property from E andF under some conditions. In Section 4, we will also show the stability of Mazur's property under the formation of Schauder decompositions and some unconditional sums of Banach spaces.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 33 , Issue 1 , January 1991 , pp. 51 - 54
Copyright
Copyright © Glasgow Mathematical Journal Trust 1991

References

REFERENCES

1.Batt, J. and Hiermeyer, W., On compactness in Lp (μ, X) in the weak topology and in the topology σ(Lp (μ, X), Lq(μ, X'), Math. Z. 82 (1983), 409–423.Google Scholar
2.Diestel, J. and Uhl, J. J. Jr, Vector measures, Math. Surveys no. 15 (American Math. Soc., 1977).CrossRefGoogle Scholar
3.Drewnowski, L., On Banach spaces with the Gelfand-Phillips property, Math. Z. 193 (1986), 405–411.CrossRefGoogle Scholar
4.Edgar, G., Measurability in a Banach space, II, Indiana Univ. Math. J. 28 (1979), 559–579.CrossRefGoogle Scholar
5.Kappeler, T., Banach spaces with the condition of Mazur, Math. Z. 191 (1986), 623–631.CrossRefGoogle Scholar
6.Schaefer, H. H., Topological vector spaces, Graduate Texts in Mathematics no. 3 (Springer 1971).CrossRefGoogle Scholar
7.Schaefer, H. H.,Banach lattices and positive operators, Die Grundlehren der Mathematischen Wissenschaften no. 215 (Springer, 1974).CrossRefGoogle Scholar
8.Schlumprecht, T., Limited sets in Banach spaces (Dissertation, Ludwig-Maximilians-Universitat, 1987).Google Scholar
9.Wilansky, A., Mazur spaces, Intemat. J. Math. Sci. 4 (1981), 39–53.CrossRefGoogle Scholar