Real and complex analysis / Walter Rudin.
Series McGraw-Hill series in higher mathematicsEditor: New York : McGraw-Hill, c1966Descripción: xi, 412 p. ; 24 cmTema(s): Mathematical analysisOtra clasificación: 00A05 (26-01 30-01 42-01 46-01)The Exponential Function, [1] Chapter 1 | Abstract Integration, [5] Set-theoretic notations and terminology, [6] The concept of measurability, [8] Simple functions, [15] Elementary properties of measures, [16] Arithmetic in [0, ∞], [18] Integration of positive functions, [19] Integration of complex functions, [24] The role played by sets of measure zero, [26] Exercises, [31] Chapter 2 | Positive Borel Measures, [33] Vector spaces, [33] Topological preliminaries, [35] The Riesz representation theorem, [40] Regularity properties of Borel measures, [47] Lebesgue measure, [49] Continuity properties of measurable functions, [53] Exercises, [56] Chapter 3 | Lp-Spaces, [60] Convex functions and inequalities, [60] The Lp-spaces, [64] Approximation by continuous functions, [68] Exercises, [70] Chapter 4 | Elementary Hilbert Space Theory, [75] Inner products and linear functionals, [75] Orthonormal sets, [81] Trigonometric series, [88] Exercises, [92] Chapter 5 | Examples of Banach Space Techniques, [95] Banach spaces, [95] Consequences of Baire’s theorem, [97] Fourier series of continuous functions, [101] Fourier coefficients of L1- functions, [103] The Hahn-Banach theorem, [105] An abstract approach to the Poisson integral, [109] Exercises, [112] Chapter [6] Complex Measures, [117] Total variation, [117] Absolute continuity, [121] Consequences of the Radon-Nikodym theorem, [126] Bounded linear functionals on Lp, [127] The Riesz representation theorem, [130] Exercises, [133] Chapter 7 | Integration on Product Spaces, [136] Measurability on cartesian products, [136] Product measures, [138] The Fubini theorem, [140] Completion of product measures, [143] Convolutions, [146] Exercises, [148] Chapter [8] Differentiation, [151] Derivatives of measures, [151] Functions of bounded variation, [160] Differentiation of point functions, [165] Differentiable transformations, [169] Exercises, [175] Chapter 9 | Fourier Transforms, [180] Formal properties, [180] The inversion theorem, [182] The Plancherel theorem, [187] The Banach algebra L1, [192] Exercises, [195] Chapter 10 | Elementary Properties of Holomorphic Functions, [198] Complex differentiation, [198] Integration over paths, [202] The Cauchy theorem, [206] The power series representation, [209] The open mapping theorem, [214] Exercises, [219] Chapter 11 | Harmonic Functions, [222] The Cauchy-Riemann equations, [222] The Poisson integral, [223] The mean value property, [230] Positive harmonic functions, [232] Exercises, [236] Chapter 12 | The Maximum Modulus Principle, [240] Introduction, [240] The Schwarz lemma, [240] The Phragmen-Lindelöf method, [243] An interpolation theorem, [246] A converse of the maximum modulus theorem, [248] Exercises, [249] Chapter 13 | Approximation by Rational Functions, [252] Preparation, [252] Runge’s theorem, [255] Cauchy’s theorem, [259] Simply connected regions, [262] Exercises, [265] Chapter 14 | Conformal Mapping, [268] Preservation of angles, [268] Linear fractional transformations, [269] Normal families, [271] The Riemann mapping theorem, [273] The class S, [276] Continuity at the boundary, [279] Conformal mapping of an annulus, [282] Exercises, [284] Chapter 15 | Zeros of Holomorphic Functions, [290] Infinite products, [290] The Weierstrass factorization theorem, [293] The Mittag-Leffler theorem, [296] Jensen’s formula, [299] Blaschke products, [302] The Muntz-Szasz theorem, [304] Exercises, [307] Chapter [16] Analytic Continuation, [312] Regular points and singular points, [312] Continuation along curves, [316] The monodromy theorem, [319] Construction of a modular function, [320] The Picard theorem, 324 Exercises, [325] Chapter 17 | Hp-Spaces, [328] Subharmonic functions, [328] The spaces and Hp, [330] The space H2, [332] The theorem of F. and M. Riesz, [335] Factorization theorems, [336] The shift operator, [341] Conjugate functions, [345] Exercises, [347] Chapter 18 | Elementary Theory of Banach Algebras, [351] Introduction, [351] The invertible elements, [352] Ideals and homomorphisms, [357] Applications, [360] Exercises, [364] Chapter 19 | Holomorphic Fourier Transforms, [367] Introduction, [367] Two theorems of Paley and Wiener, [368] Quasi-analytic classes, [372] The Denjoy-Carleman theorem, [376] Exercises, [379] Chapter 20 | Uniform Approximation by Polynomials, [382] Introduction, [382] Some lemmas, [383] Mergelyan’s theorem, [386] Exercises, [390] Appendix | Hausdorff’s Maximality Theorem, [391] Notes and Comments, [393] Bibliography, [401] List of Special Symbols, [403] Index, [405]
| Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
|---|---|---|---|---|---|---|---|---|
Libros
|
Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 00A05 R916 (Browse shelf) | Available | A-2536 |
Bibliografía: p. 401-402.
The Exponential Function, [1] --
Chapter 1 | Abstract Integration, [5] --
Set-theoretic notations and terminology, [6] --
The concept of measurability, [8] --
Simple functions, [15] --
Elementary properties of measures, [16] --
Arithmetic in [0, ∞], [18] --
Integration of positive functions, [19] --
Integration of complex functions, [24] --
The role played by sets of measure zero, [26] --
Exercises, [31] --
Chapter 2 | Positive Borel Measures, [33] --
Vector spaces, [33] --
Topological preliminaries, [35] --
The Riesz representation theorem, [40] --
Regularity properties of Borel measures, [47] --
Lebesgue measure, [49] --
Continuity properties of measurable functions, [53] --
Exercises, [56] --
Chapter 3 | Lp-Spaces, [60] --
Convex functions and inequalities, [60] --
The Lp-spaces, [64] --
Approximation by continuous functions, [68] --
Exercises, [70] --
Chapter 4 | Elementary Hilbert Space Theory, [75] --
Inner products and linear functionals, [75] --
Orthonormal sets, [81] --
Trigonometric series, [88] --
Exercises, [92] --
Chapter 5 | Examples of Banach Space Techniques, [95] --
Banach spaces, [95] --
Consequences of Baire’s theorem, [97] --
Fourier series of continuous functions, [101] --
Fourier coefficients of L1- functions, [103] --
The Hahn-Banach theorem, [105] --
An abstract approach to the Poisson integral, [109] --
Exercises, [112] --
Chapter [6] --
Complex Measures, [117] --
Total variation, [117] --
Absolute continuity, [121] --
Consequences of the Radon-Nikodym theorem, [126] --
Bounded linear functionals on Lp, [127] --
The Riesz representation theorem, [130] --
Exercises, [133] --
Chapter 7 | Integration on Product Spaces, [136] --
Measurability on cartesian products, [136] --
Product measures, [138] --
The Fubini theorem, [140] --
Completion of product measures, [143] --
Convolutions, [146] --
Exercises, [148] --
Chapter [8] --
Differentiation, [151] --
Derivatives of measures, [151] --
Functions of bounded variation, [160] --
Differentiation of point functions, [165] --
Differentiable transformations, [169] --
Exercises, [175] --
Chapter 9 | Fourier Transforms, [180] --
Formal properties, [180] --
The inversion theorem, [182] --
The Plancherel theorem, [187] --
The Banach algebra L1, [192] --
Exercises, [195] --
Chapter 10 | Elementary Properties of Holomorphic Functions, [198] --
Complex differentiation, [198] --
Integration over paths, [202] --
The Cauchy theorem, [206] --
The power series representation, [209] --
The open mapping theorem, [214] --
Exercises, [219] --
Chapter 11 | Harmonic Functions, [222] --
The Cauchy-Riemann equations, [222] --
The Poisson integral, [223] --
The mean value property, [230] --
Positive harmonic functions, [232] --
Exercises, [236] --
Chapter 12 | The Maximum Modulus Principle, [240] --
Introduction, [240] --
The Schwarz lemma, [240] --
The Phragmen-Lindelöf method, [243] --
An interpolation theorem, [246] --
A converse of the maximum modulus theorem, [248] --
Exercises, [249] --
Chapter 13 | Approximation by Rational Functions, [252] --
Preparation, [252] --
Runge’s theorem, [255] --
Cauchy’s theorem, [259] --
Simply connected regions, [262] --
Exercises, [265] --
Chapter 14 | Conformal Mapping, [268] --
Preservation of angles, [268] --
Linear fractional transformations, [269] --
Normal families, [271] --
The Riemann mapping theorem, [273] --
The class S, [276] --
Continuity at the boundary, [279] --
Conformal mapping of an annulus, [282] --
Exercises, [284] --
Chapter 15 | Zeros of Holomorphic Functions, [290] --
Infinite products, [290] --
The Weierstrass factorization theorem, [293] --
The Mittag-Leffler theorem, [296] --
Jensen’s formula, [299] --
Blaschke products, [302] --
The Muntz-Szasz theorem, [304] --
Exercises, [307] --
Chapter [16] --
Analytic Continuation, [312] --
Regular points and singular points, [312] --
Continuation along curves, [316] --
The monodromy theorem, [319] --
Construction of a modular function, [320] --
The Picard theorem, 324 Exercises, [325] --
Chapter 17 | Hp-Spaces, [328] --
Subharmonic functions, [328] --
The spaces and Hp, [330] --
The space H2, [332] --
The theorem of F. and M. Riesz, [335] --
Factorization theorems, [336] --
The shift operator, [341] --
Conjugate functions, [345] --
Exercises, [347] --
Chapter 18 | Elementary Theory of Banach Algebras, [351] --
Introduction, [351] --
The invertible elements, [352] --
Ideals and homomorphisms, [357] --
Applications, [360] --
Exercises, [364] --
Chapter 19 | Holomorphic Fourier Transforms, [367] --
Introduction, [367] --
Two theorems of Paley and Wiener, [368] --
Quasi-analytic classes, [372] --
The Denjoy-Carleman theorem, [376] --
Exercises, [379] --
Chapter 20 | Uniform Approximation by Polynomials, [382] --
Introduction, [382] --
Some lemmas, [383] --
Mergelyan’s theorem, [386] --
Exercises, [390] --
Appendix | Hausdorff’s Maximality Theorem, [391] --
Notes and Comments, [393] --
Bibliography, [401] --
List of Special Symbols, [403] --
Index, [405] --
MR, 35 #1420
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