Handbook of the geometry of Banach spaces. Volume 1 (Q2743133)

The articles of this volume will be reviewed individually.NEWLINENEWLINENEWLINE Publishers's description: The Handbook presents an overview of most aspects of modern Banach space theory and its applications. The up-to-date surveys, authored by leading research workers in the area, are written to be accessible to a wide audience. In addition to presenting the state of the art of Banach space theory, the surveys discuss the relation of the subject with such areas as harmonic analysis, complex analysis, classical convexity, probability theory, operator theory, combinatorics, logic, geometric measure theory, and partial differential equations. The Handbook begins with a chapter on basic concepts in Banach space theory which contains all the background needed for reading any other chapter in the Handbook. Each of the twenty one articles in this volume after the basic concepts chapter is devoted to one specific direction of Banach space theory or its applications. Each article contains a motivated introduction as well as an exposition of the main results, methods, and open problems in its specific direction. Most have an extensive bibliography. Many articles contain new proofs of known results as well as expositions of proofs which are hard to locate in the literature or are only outlined in the original research papers. As well as being valuable to experienced researchers in Banach space theory, the Handbook should be an outstanding source for inspiration and information to graduate students and beginning researchers. The Handbook will be useful for mathematicians who want to get an idea of the various developments in Banach space theory.NEWLINENEWLINEIndexed articles:NEWLINENEWLINE\textit{Johnson, William B.; Lindenstrauss, Joram}, Basic concepts in the geometry of Banach spaces, 1-84 [Zbl 1011.46009]NEWLINENEWLINE\textit{Abramovich, Y. A.; Aliprantis, C. D.}, Positive operators, 85-122 [Zbl 1202.47042]NEWLINENEWLINE\textit{Alspach, Dale; Odell, Edward}, \(L_p\) spaces, 123-159 [Zbl 1008.46013]NEWLINENEWLINE\textit{Ball, Keith}, Convex geometry and functional analysis, 161-194 [Zbl 1017.46004]NEWLINENEWLINE\textit{Bourgain, Jean}, \(\Lambda_p\)-sets in analysis: Results, problems and related aspects, 195-232 [Zbl 1016.43004]NEWLINENEWLINE\textit{Burkholder, Donald L.}, Martingales and singular integrals in Banach spaces, 233-269 [Zbl 1029.46007]NEWLINENEWLINE\textit{Casazza, Peter G.}, Approximation properties., 271-316 [Zbl 1067.46025]NEWLINENEWLINE\textit{Davidson, Kenneth R.; Szarek, Stanislaw J.}, Local operator theory, random matrices and Banach spaces., 317-366 [Zbl 1067.46008]NEWLINENEWLINE\textit{Delbaen, Freddy; Schachermayer, Walter}, Applications to mathematical finance, 367-391 [Zbl 1013.46062]NEWLINENEWLINE\textit{Deville, Robert; Ghoussoub, Nassif}, Perturbed minimization principles and applications., 393-435 [Zbl 1040.46011]NEWLINENEWLINE\textit{Diestel, Joe; Jarchow, Hans; Pietsch, Albrecht}, Operator ideals, 437-496 [Zbl 1012.47001]NEWLINENEWLINE\textit{Dilworth, S. J.}, Special Banach lattices and their applications., 497-532 [Zbl 1042.46013]NEWLINENEWLINE\textit{Enflo, P.; Lomonosov, V.}, Some aspects of the invariant subspace problem, 533-559 [Zbl 1026.47004]NEWLINENEWLINE\textit{Figiel, T.; Wojtaszczyk, P.}, Special bases in function spaces, 561-597 [Zbl 1018.46007]NEWLINENEWLINE\textit{Fonf, V. P.; Lindenstrauss, J.; Phelps, R. R.}, Infinite dimensional convexity., 599-670 [Zbl 1086.46004]NEWLINENEWLINE\textit{Gamelin, T. W.; Kislyakov, S. V.}, Uniform algebras as Banach spaces, 671-706 [Zbl 1032.46067]NEWLINENEWLINE\textit{Giannopoulos, Apostolos A.; Milman, Vitali D.}, Euclidean structure in finite dimensional normed spaces, 707-779 [Zbl 1009.46004]NEWLINENEWLINE\textit{Godefroy, Gilles}, Renormings of Banach spaces, 781-835 [Zbl 1009.46003]NEWLINENEWLINE\textit{Johnson, William B.; Schechtman, Gideon}, Finite dimensional subspaces of \(L_p\), 837-870 [Zbl 1012.46012]NEWLINENEWLINE\textit{Kislyakov, S. V.}, Banach spaces and classical harmonic analysis, 871-898 [Zbl 1024.46002]NEWLINENEWLINE\textit{Koldobsky, Alexander; König, Hermann}, Aspects of the isometric theory of Banach spaces, 899-939 [Zbl 1005.46005]NEWLINENEWLINE\textit{König, Hermann}, Eigenvalues of operators and applications, 941-974 [Zbl 1006.47022]