Complex Analysis in Banach Spaces

Overview


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. 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 ...
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Overview


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. 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 Fr├ęchet 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.
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Product Details

  • ISBN-13: 9780486474663
  • Publisher: Dover Publications
  • Publication date: 3/18/2010
  • Series: Dover Books on Mathematics Series
  • Pages: 464
  • Sales rank: 1,435,202
  • Product dimensions: 6.20 (w) x 9.20 (h) x 1.00 (d)

Table of Contents

Chapter I Polynomials

1 Multilinear mappings 1

2 Polynomials 12

3 Polynomials of one variable 18

4 Power series 27

Chapter II Holomorphic Mappings

5 Holomorphic mappings 33

6 Vector-valued integration 40

7 The Cauchy integral formulas 45

8 G-holomorphic mappings 58

9 The compact-open topology 69

Chapter III Domains of Holomorphy

10 Domains of holomorphy 79

11 Holomorphically convex domains 85

12 Bounding sets 94

Chapter IV Differentiable Mappings

13 Differentiable mappings 99

14 Differentiable mappings of higher order 111

15 Partitions of unity 118

16 Test functions 122

17 Distributions 127

Chapter V Differential Forms

18 Alternating multilinear forms 139

19 Differential forms 144

20 The Poincar? lemma 153

21 The &?bbar; operator 156

22 Differential forms with bounded support 162

23 The &?bbar; equation in polydiscs 168

Chapter VI Polynomially Convex Domains

24 Polynomially convex compact sets in &cslash;n 177

25 Polynomially convex domains in &cslash;n 185

26 Schauder bases 188

27 The approximation property 194

28 Polynomial approximation in Banach spaces 202

Chapter VII Commutative Banach Algebras

29 Banach algebras 211

30 Commutative Banach algebras 214

31 The joint spectrum 219

32 Projective limits of Banach algebras 227

33 The Michael problem 236

Chapter VIII Plurisubharmonic Functions

34 Plurisubharmonic functions 245

35 Regularization of plurisubharmonic functions 255

36 Separately holomorphic mappings 265

37 Pseudoconvex domains 273

38 Plurisubharmonic functions on pseudoconvex domains 279

Chapter IX The &?bbar; Equation in Pseudoconvex Domains

39 Densely defined operators in Hilbert spaces 287

40 The &?bbar; operator for L2 differential forms 291

41 L2 solutions of the &?bbar; equation 300

42 C? solutions of the &?bbar; equation 307

Chapter X The Levi Problem

43 The Levi problem in &Cslash;n 311

44 Holomorphic approximation in &Cslash;n 313

45 The Levi problem in Banach spaces 320

46 Holomorphic approximation in Banach spaces 325

Chapter XI Riemann Domains

47 Riemann domains 331

48 Distributions on Riemann domains 339

49 Pseudoconvex Riemann domains 346

50 Plurisubharmonic functions on Riemann domains 353

51 The &?bbar; equation in Riemann domains 359

Chapter XII The Levi Problem in Riemann Domains

52 The Cartan-Thullen theorem in Riemann domains 361

53 The Levi problem in finite dimensional Riemann domains 372

54 The Levi problem in infinite dimensional Riemann domains 380

55 Holomorphic approximation in infinite dimensional Riemann domains 391

Chapter XIII Envelopes of Holomorphy

56 Envelopes of holomorphy 397

57 The spectrum 400

58 Envelopes of holomorphy and the spectrum 408

Bibliography 421

Index 431

Errata 435

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