zbMATH — the first resource for mathematics

Banach spaces of vector-valued functions. (English) Zbl 0902.46017
Lecture Notes in Mathematics. 1676. Berlin: Springer. viii, 118 p. (1997).
This small monograph is devoted to the classical problems of geometry of Banach spaces about the existence of copies and complemented copies of \(c_0\), \(\ell_1\) and \(\ell_\infty\) in Banach vector-function spaces \(L_p(\mu,X)\) \((1\leq p\leq\infty)\) and \({\mathcal C}(K,X)\) (here \(X\) is a Banach space) under different assumptions about \(\mu\) and \(K\). The authors present an exhausting and systematical summary of all, known to this time, results in this field (in particular, results by S. Kwapień, G. Pisier, E. Saab – P. Saab, C. Cembranos, F. Freniche, L. Drewnowski, F. Bombal, G. Emmanuelle, J. Mendoza, D. Leuing – F. Räbiger, S. Díaz, N. J. Kalton and others) and formulate some unsolved problems. In particular, the authors describe all situations when the existence of copies or complemented copies of \(c_0\), \(\ell_1\) and \(\ell_\infty\) in spaces \(L_p(\mu, X)\) \((1\leq p\leq\infty)\) and in \({\mathcal C}(K, X)\) is turned out to be equivalent to their existence in the spaces \(X\) or \(L_p(\mu)\) and \({\mathcal C}(K)\) correspondingly. The book is written in laconic and clear manner. It can be considered as a good supplement to the classical books in the field of J. Diestel, J. Diestel and J. J. Uhl. Undoubtedly, this book will be useful for all specialists in the field.
Reviewer: P.Zabreiko (Minsk)

46E40 Spaces of vector- and operator-valued functions
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
46B25 Classical Banach spaces in the general theory
Full Text: DOI