Analyticity in Infinite Dimensional Spaces

✍ Scribed by Michel Hervé

• Publisher: De Gruyter
• Year: 1989
• Tongue: English
• Leaves: 216
• Series: De Gruyter Studies in Mathematics; 10
• Edition: Reprint 2011
• Category: Library

✦ Synopsis

"Many excellent books in this field have been published. But this is the first book that describes systematically the former, although this object is very important and very interesting. And in the latter also this book contains a number of results which are not touched on in other books. Therefore the reviewer is sure that many results contained in it will be of great interest to readers." Mathematical Reviews

"This book is a well written introduction to the study of these questions. A large number of examples are given to illustrate the different phenomena studied and to show the necessity of the hypotheses in the theorems." Zentralblatt für Mathematik

✦ Table of Contents

Chapter 1 Some topological preliminaries
Summary
1.1 Locally convex spaces
1.2 Vector valued infinite sums and integrals
1.3 Baire spaces
1.4 Barrelled spaces
1.5 Inductive limits
Chapter 2 Gâteaux-analyticity
Summary
2.1 Vector valued functions of several complex variables
2.2 Polynomials and polynomial maps
2.3 Gâteaux-analyticity
2.4 Boundedness and continuity of Gâteaux-analytic maps
Exercises
Chapter 3 Analyticity, or Fréchet-analyticity
Summary
3.1 Equivalent definitions
3.2 Separate analyticity
3.3 Entire maps and functions
3.4 Bounding sets
Exercises
Chapter 4 Plurisubharmonic functions
Summary
4.1 Plurisubharmonic functions on an open set Ω in a I.c. space X
4.2 The finite dimensional case
4.3 Back to the infinite dimensional case
4.4 Analytic maps and pluriharmonic functions
4.5 Polar subsets
4.6 A fine maximum principle
Exercises
Chapter 5 Problems involving plurisubharmonic functions
Summary
5.1 Pseudoconvexity in a I.c. space X
5.2 The Levi problem
5.3 Boundedness of p.s.h. functions and entire maps
5.4 The growth of p.s.h. functions and entire maps
5.5 The density number for a p.s.h. function
Exercises
Chapter 6 Analytic maps from a given domain to another one
Summary
6.1 A generalization of the Lindelöf principle
6.2 Intrinsic pseudodistances
6.3 Complex geodesics and complex extremal points
6.4 Automorphisms and fixed points
Exercises
Bibliography
Glossary of Notations
Subject Index