Complex Analysis in Banach Spaces : Holomorphic Functions and Domains of Holomorphy in Finite and Infinite Dimensions

Main subject categories: • Holomorphic functions • Domains of holomorphy • Banach spaces • Complex analysisThe development of complex analysis is based on issues related to holomorphic continuation and holomorphic approximation. This volume presents a unified view of these topics in finite and infinite dimensions. A high-level tutorial in pure and applied mathematics, its prerequisites include a familiarity with the basic properties of holomorphic functions, the principles of Banach and Hilbert spaces, and the theory of Lebesgue integration.The four-part treatment begins with an overview of the basic properties of holomorphic mappings and holomorphic domains in Banach spaces. The second section explores differentiable mappings, differentiable forms, and polynomially convex compact sets, in which the results are applied to the study of Banach and Fréchet algebras. Subsequent sections examine plurisubharmonic functions and pseudoconvex domains in Banach spaces, along with Riemann domains and envelopes of holomorphy. In addition to its value as a text for advanced graduate students of mathematics, this volume also functions as a reference for researchers and professionals. Content: Editor Page ii Edited by Page iii Copyright page Page iv Dedication Page v foreword Pages vii-viii Jorge Mujica Chapter I Polynomials Pages 1-32 Chapter II Holomorphic Mappings Pages 33-77 Chapter III Domains of Holomorphy Pages 79-97 Chapter IV Differentiable Mappings Pages 99-138 Chapter V Differential Forms Pages 139-176 Chapter VI Polynomially Convex Domains Pages 177-210 Chapter VII COMMUTATIVE BANACH ALGEBRAS Pages 211-244 Chapter VIII Plurisubharmonic Functions Pages 245-286 Chapter IX The ∂ Equation in Pseudoconvex Domains Pages 287-310 Chapter X The Levi Problem Pages 311-330 Chapter XI RIEMANN DOMAINS Pages 331-360 Chapter XII The Levi Problem in Riemann Domains Pages 361-395 Chapter XIII Envelopes of Holomorphy Pages 397-420 Birliography Pages 421-429 Index Pages 431-434 Problems arising from the study of holomorphic continuation and holomorphic approximation have been central in the development of complex analysis in finitely many variables, and constitute one of the most promising lines of research in infinite dimensional complex analysis. This book presents a unified view of these topics in both finite and infinite dimensions.

The development of complex analysis is based on issues related to holomorphic continuation and holomorphic approximation. This volume presents a unified view of these topics in finite and infinite dimensions. In addition to its value as a text for graduate students of mathematics, it functions as a reference for researchers and professionals. 1986 edition.