Banach spaces of vector-valued functions
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.
scientific article; zbMATH DE number 1095902