دانلود کتاب Analyticity in Infinite Dimensional Spaces

During the last twenty years, the theory of analyticity in infinite dimensions has developed from its foundations into a structure which may be termed harmonious, provided that one accepts to do without some features of the finite dimensional case. This harmony is of course favoured by the choice of a unique setting -locally convex spaces over the complex field, analytic maps into sequentially complete spaces - and a central topic: plurisub- harmonicity, where a multitude of results obtained by different authors - Pierre Lelong, the founder of the notion, Gerard Coeure, Christer Kiselman and others - deserved to be brought together. The reader will find the precise contents of each chapter in the summary which opens it. The concern for unity has inevitably led to the omission of several other topics in spite of their indisputable interest, and among th(:se I insist on the local theory of analytic sets. But the methods used have little in common with those used in this book; the material consists essentially in two theses: "Sous-ensembles analytiques d'une variete analytique banachique" (Paris, 1969) by Jean-Pierre Ramis, "Ensembles analytiques complexes dans les espaces localement convexes" (Paris, 1969) by Pierre Mazet, and for the time being there is little more to say on the subject. This book is a tribute to all the authors it mentions. The first chapters owe much to the paper [Boc Sic]; comparatively short sections are devoted to some topics extensively developed in the excellent monographs pub- lished as North Holland mathematics studies: "Pseudo-convexite, con- vexite polynomiale et domaines d'holomorphie" by Philippe Noverraz (n° 3); "Analytic functions and manifolds in infinite dimensional spaces" by Gerard Coeure (n° 11); "Holomorphic maps and invariant distances" by Franzoni and Vesentini (n° 40); "Complex analysis in locally com- plex spaces" by Sean Dineen (n° 57); "Complex analysis in Banach spaces" by Jorge Mujica (n° 120). To bring a new contribution was no easy task, but it seemed to me that classical potential theory deserved a better place, as a special tribute to the recently deceased Marcel Brelot. Finally I express my gratitude to Professors Heinz Bauer and Peter Gabriel who accepted this book in their renowned series. Contents:Chapter 1 Some topological preliminaries . . . . . . . . . . . . . . . . . . . . . 1 Summary..... . .. .. ....... .. ......... . ........ .. . ... ... 1 1.1 Locally convex spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Vector valued infinite sums and integrals. . . . . . . . . . . . . . . . 6 1.3 Baire spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Barrelled spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5 Inductive limits .................................. . . . 13 Chapter 2 Gateaux-analyticity.............................. 19 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1 Vector valued functions of several complex variables . . . . . . 20 2.2 Polynomials and polynomial maps . . . . . . . . . . . . . . . . . . . . . 28 2.3 Gateaux-analyticity............................... . . . 35 2.4 Boundedness and continuity of Gateaux-analytic maps. . . . 43 Exercises ........................................... . . . 50 Chapter 3 Analyticity, or Frechet-analyticity . . . . . . . . . . . . . . . . . . 51 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 51 3.1 Equivalent definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2 Separate analyticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3 Entire maps and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.4 Bounding sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73 Exercises ........................................... . . . 79 Chapter 4 Plurisubharmonic functions. . . . . . . . . . . . . . . . . . . . . . . . 81 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.1 Plurisubharmonic functions on an open set n in a l.c. space X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82 4.2 The finite dimensional case ........................... 87 4.3 Back to the infinite dimensional case ................... 94 4.4 Analytic maps and pluriharmonic functions ............. 104 4.5 Polar subsets ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.6 A fine maximum principle ............................ 120 Exercises .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Chapter 5 Problems involving plurisubharmonic functions. . . . . . . . 129 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.1 Pseudoconvexity in a 1.c. space X ...................... 130 5.2 The Levi problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.3 Boundedness of p.s.h. functions and entire maps. . . . . . . . . . 144 5.4 The growth of p.s.h. functions and entire maps. . . . . . . . . . . 146 5.5 The density number for a p.s.h. function. . . . . . . . . . . . . . . . . 154 Exercises .............................................. 162 Chapter 6 Analytic maps from a given domain to another one. . . . . 163 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6.1 A generalization of the Lindelof principle. . . . . . . . . . . . . . . . 164 6.2 Intrinsic pseudodistances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 6.3 Complex geodesics and complex extremal points ......... 179 6.4 Automorphisms and fixed points. . . . . . . . . . . . . . . . . . . . . . . 184 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 194 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 195 Glossary of Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .201 Subject Index ............................................ 205