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Elements of Functional Analysis

Francis Hirsch, Gilles Lacombe (Autoren)

Buch | Hardcover

396 Seiten

1999 | 1999 ed.
Springer-Verlag New York Inc.
978-0-387-98524-4 (ISBN)

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This book presents the fundamental function spaces and their duals, explores operator theory and finally develops the theory of distributions up to significant applications such as Sobolev spaces and Dirichlet problems. Includes an assortment of well formulated exercises, with answers and hints collected at the end of the book.

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.

Reihe/Serie	Graduate Texts in Mathematics ; 192
Übersetzer	S. Levy
Zusatzinfo	XIV, 396 p.
Verlagsort	New York, NY
Sprache	englisch
Maße	155 x 235 mm
Gewicht	1690 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
ISBN-10	0-387-98524-7 / 0387985247
ISBN-13	978-0-387-98524-4 / 9780387985244
Zustand	Neuware