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A survey of results related to the Dunford-Pettis property. (English) Zbl 0571.46013
Integration, topology, and geometry in linear spaces, Proc. Conf., Chapel Hill/N.C. 1979, Contemp. Math. 2, 15-60 (1980).
[For the entire collection see Zbl 0541.00009.]
A Banach space X is said to have the Dunford-Pettis property (DPP for short) if every weakly compact operator from X to any other Banach space is completely continuous, i.e. takes weakly compact sets onto norm compact sets. This notion originates from a classical result of N. Dunford and B. J. Pettis [Trans. Am. Math. Soc. 47, 323-392 (1940; Zbl 0023.32902)] which states that \(L^ 1\)-spaces have DPP.
Diestel’s survey provides the reader with a thorough and interesting introduction to nearly everything which was known about DPP at the end of the seventies.
In the meantime the subject has become an indispensable part of Banach space theory the more accessible aspects of which can also to be found in a series of exercises in the monographs of B. Beauzamy [”Introduction to Banach spaces and their geometry” (1982; Zbl 0491.46014)] or J. Diestel [”Sequences and Series in Banach spaces” (1984; Zbl 0542.46007)]. As the most important contribution to DPP after the appearance of the paper under reference the reader is referred to J. Bourgain: Proc. Am. Math. Soc. 81, 265-272 (1981; Zbl 0463.46027).
Reviewer: E.Behrends

MSC:
46B20 Geometry and structure of normed linear spaces
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
47B38 Linear operators on function spaces (general)