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This is a graduate text on functional analysis. After presenting the fundamental function spaces and their duals, the authors study topics in operator theory and finally develop the theory of distributions up to significant applications such as Sobolev spaces and Dirichlet problems.Along the way, the reader is presented with a truly remarkable assortment of well formulated and interesting exercises, which test the understanding as well as point out many related topics. The answers and hints that are not already contained in the statements of the exercises are collected at the end of the book.
This is a graduate text on functional analysis. After presenting the fundamental function spaces and their duals, the authors study topics in operator theory and finally develop the theory of distributions up to significant applications such as Sobolev spaces and Dirichlet problems.Along the way, the reader is presented with a truly remarkable assortment of well formulated and interesting exercises, which test the understanding as well as point out many related topics. The answers and hints that are not already contained in the statements of the exercises are collected at the end of the book.
Produktdetails
- Produktdetails
- Graduate Texts in Mathematics 192
- Verlag: Springer / Springer New York / Springer, Berlin
- 1999
- Seitenzahl: 396
- Englisch
- Abmessung: 275mm x 159mm x 29mm
- Gewicht: 714g
- ISBN-13: 9780387985244
- ISBN-10: 0387985247
- Artikelnr.: 09230467
- Herstellerkennzeichnung Die Herstellerinformationen sind derzeit nicht verfügbar.
- Graduate Texts in Mathematics 192
- Verlag: Springer / Springer New York / Springer, Berlin
- 1999
- Seitenzahl: 396
- Englisch
- Abmessung: 275mm x 159mm x 29mm
- Gewicht: 714g
- ISBN-13: 9780387985244
- ISBN-10: 0387985247
- Artikelnr.: 09230467
- Herstellerkennzeichnung Die Herstellerinformationen sind derzeit nicht verfügbar.
Prologue: Sequences.- 1 Countability.- 2 Separability.- 3 The Diagonal Procedure.- 4 Bounded Sequences of Continuous Linear Maps.- I Function Spaces and Their Duals.- 1 The Space of Continuous Functions on a Compact Set.- 1 Generalities.- 2 The Stone-Weierstrass Theorems.- 3 Ascoli's Theorem.- 2 Locally Compact Spaces and Radon Measures.- 1 Locally Compact Spaces.- 2 Daniell's Theorem.- 3 Positive Radon Measures.- 3A Positive Radon Measures on $${{mathbb{R}}^{d}}$$ and the Stieltjes Integral.- 3B Surface Measure on Spheres in $${{mathbb{R}}^{d}}$$.- 4 Real and Complex Radon Measures.- 3 Hilbert Spaces.- 1 Definitions, Elementary Properties, Examples.- 2 The Projection Theorem.- 3 The Riesz Representation Theorem.- 3A Continuous Linear Operators on a Hilbert Space.- 3B Weak Convergence in a Hilbert Space.- 4 Hilbert Bases.- 4 LpSpaces.- 1 Definitions and General Properties.- 2 Duality.- 3 Convolution.- II Operators.- 5 Spectra.- 1 Operators on Banach Spaces.- 2 Operators in Hilbert Spaces.- 2A Spectral Properties of Hermitian Operators.- 2B Operational Calculus on Hermitian Operators.- 6 Compact Operators.- 1 General Properties.- lA Spectral Properties of Compact Operators.- 2 Compact Selfadjoint Operators.- 2A Operational Calculus and the Fredholm Equation.- 2B Kernel Operators.- III Distributions.- 7 Definitions and Examples.- 1 Test Functions.- lA Notation.- 1B Convergence in Function Spaces.- 1C Smoothing.- 1D C?Partitions of Unity.- 2 Distributions.- 2A Definitions.- 2B First Examples.- 2C Restriction and Extension of a Distribution to an Open Set.- 2D Convergence of Sequences of Distributions.- 2E Principal Values.- 2F Finite Parts.- 3 Complements.- 3A Distributions of Finite Order.- 3B The Support of a Distribution.- 3C Distributions with Compact Support.- 8 Multiplication and Differentiation.- 1 Multiplication.- 2 Differentiation.- 3 Fundamental Solutions of a Differential Operator.- 3A The Laplacian.- 3B The Heat Operator.- 3C The Cauchy-Riemann Operator.- 9 Convolution of Distributions.- 1 Tensor Product of Distributions.- 2 Convolution of Distributions.- 2A Convolution in ??.- 2B Convolution in D?.- 2C Convolution of a Distribution with a Function.- 3 Applications.- 3A Primitives and Sobolev's Theorem.- 3B Regularity.- 3C Fundamental Solutions and Partial Differential Equations.- 3D The Algebra D+?.- 10 The Laplacian on an Open Set.- 1 The spaces H1(?) and H01(?).- 2 The Dirichlet Problem.- 2A The Dirichlet Problem.- 2B The Heat Problem.- 2C The Wave Problem.- Answers to the Exercises.
Prologue: Sequences.- 1 Countability.- 2 Separability.- 3 The Diagonal Procedure.- 4 Bounded Sequences of Continuous Linear Maps.- I Function Spaces and Their Duals.- 1 The Space of Continuous Functions on a Compact Set.- 1 Generalities.- 2 The Stone-Weierstrass Theorems.- 3 Ascoli's Theorem.- 2 Locally Compact Spaces and Radon Measures.- 1 Locally Compact Spaces.- 2 Daniell's Theorem.- 3 Positive Radon Measures.- 3A Positive Radon Measures on $${{mathbb{R}}^{d}}$$ and the Stieltjes Integral.- 3B Surface Measure on Spheres in $${{mathbb{R}}^{d}}$$.- 4 Real and Complex Radon Measures.- 3 Hilbert Spaces.- 1 Definitions, Elementary Properties, Examples.- 2 The Projection Theorem.- 3 The Riesz Representation Theorem.- 3A Continuous Linear Operators on a Hilbert Space.- 3B Weak Convergence in a Hilbert Space.- 4 Hilbert Bases.- 4 LpSpaces.- 1 Definitions and General Properties.- 2 Duality.- 3 Convolution.- II Operators.- 5 Spectra.- 1 Operators on Banach Spaces.- 2 Operators in Hilbert Spaces.- 2A Spectral Properties of Hermitian Operators.- 2B Operational Calculus on Hermitian Operators.- 6 Compact Operators.- 1 General Properties.- lA Spectral Properties of Compact Operators.- 2 Compact Selfadjoint Operators.- 2A Operational Calculus and the Fredholm Equation.- 2B Kernel Operators.- III Distributions.- 7 Definitions and Examples.- 1 Test Functions.- lA Notation.- 1B Convergence in Function Spaces.- 1C Smoothing.- 1D C?Partitions of Unity.- 2 Distributions.- 2A Definitions.- 2B First Examples.- 2C Restriction and Extension of a Distribution to an Open Set.- 2D Convergence of Sequences of Distributions.- 2E Principal Values.- 2F Finite Parts.- 3 Complements.- 3A Distributions of Finite Order.- 3B The Support of a Distribution.- 3C Distributions with Compact Support.- 8 Multiplication and Differentiation.- 1 Multiplication.- 2 Differentiation.- 3 Fundamental Solutions of a Differential Operator.- 3A The Laplacian.- 3B The Heat Operator.- 3C The Cauchy-Riemann Operator.- 9 Convolution of Distributions.- 1 Tensor Product of Distributions.- 2 Convolution of Distributions.- 2A Convolution in ??.- 2B Convolution in D?.- 2C Convolution of a Distribution with a Function.- 3 Applications.- 3A Primitives and Sobolev's Theorem.- 3B Regularity.- 3C Fundamental Solutions and Partial Differential Equations.- 3D The Algebra D+?.- 10 The Laplacian on an Open Set.- 1 The spaces H1(?) and H01(?).- 2 The Dirichlet Problem.- 2A The Dirichlet Problem.- 2B The Heat Problem.- 2C The Wave Problem.- Answers to the Exercises.