Elements of Functional AnalysisSpringer Science & Business Media, 26 mars 1999 - 396 pages This book arose from a course taught for several years at the Univer sity of Evry-Val d'Essonne. It is meant primarily for graduate students in mathematics. To make it into a useful tool, appropriate to their knowl edge level, prerequisites have been reduced to a minimum: essentially, basic concepts of topology of metric spaces and in particular of normed spaces (convergence of sequences, continuity, compactness, completeness), of "ab stract" integration theory with respect to a measure (especially Lebesgue measure), and of differential calculus in several variables. The book may also help more advanced students and researchers perfect their knowledge of certain topics. The index and the relative independence of the chapters should make this type of usage easy. The important role played by exercises is one of the distinguishing fea tures of this work. The exercises are very numerous and written in detail, with hints that should allow the reader to overcome any difficulty. Answers that do not appear in the statements are collected at the end of the volume. There are also many simple application exercises to test the reader's understanding of the text, and exercises containing examples and coun terexamples, applications of the main results from the text, or digressions to introduce new concepts and present important applications. Thus the text and the exercises are intimately connected and complement each other. |
Table des matières
Sequences | 1 |
2 Separability | 7 |
3 The Diagonal Procedure | 12 |
4 Bounded Sequences of Continuous Linear Maps | 18 |
FUNCTION SPACES AND THEIR DUALS | 25 |
The Space of Continuous Functions on a Compact Set Introduction and Notation | 27 |
1 Generalities | 28 |
2 The StoneWeierstrass Theorems | 31 |
DISTRIBUTIONS | 255 |
Definitions and Examples | 257 |
IB Convergence in Function Spaces | 259 |
1C Smoothing | 261 |
ID C Partitions of Unity | 262 |
2 Distributions | 267 |
2B First Examples | 268 |
2C Restriction and Extension of a Distribution to an Open Set | 271 |
3 Ascolis Theorem | 42 |
Locally Compact Spaces and Radon Measures | 49 |
2 Daniells Theorem | 57 |
3 Positive Radon Measures | 68 |
3A Positive Radon Measures on R and the Stieltjes Integral | 71 |
3B Surface Measure on Spheres in Rd For r 0 we consider the sets | 74 |
4 Real and Complex Radon Measures | 86 |
Hilbert Spaces | 97 |
2 The Projection Theorem | 105 |
3 The Riesz Representation Theorem | 111 |
3A Continuous Linear Operators on a Hilbert Space | 112 |
3B Weak Convergence in a Hilbert Space | 114 |
4 Hilbert Bases | 123 |
LP Spaces | 143 |
2 Duality | 159 |
3 Convolution | 169 |
OPERATORS | 185 |
Spectra | 187 |
2 Operators in Hilbert Spaces | 201 |
2A Spectral Properties of Hermitian Operators | 203 |
2B Operational Calculus on Hermitian Operators | 205 |
Compact Operators | 213 |
1A Spectral Properties of Compact Operators | 217 |
2 Compact Selfadjoint Operators | 234 |
2A Operational Calculus and the Fredholm Equation | 238 |
2B Kernel Operators | 240 |
2D Convergence of Sequences of Distributions | 272 |
2F Finite Parts | 273 |
3 Complements | 280 |
3B The Support of a Distribution | 281 |
Multiplication and Differentiation | 287 |
2 Differentiation | 292 |
3 Fundamental Solutions of a Differential Operator | 306 |
3A The Laplacian | 307 |
3B The Heat Operator | 310 |
3C The CauchyRiemann Operator | 311 |
Convolution of Distributions | 317 |
2 Convolution of Distributions | 324 |
2B Convolution in Q | 325 |
2C Convolution of a Distribution with a Function | 332 |
3 Application | 337 |
3B Regularity | 340 |
3C Fundamental solutions and Partial Differential Equations | 343 |
The Laplacian on an Open Set | 349 |
2 The Dirichlet Problem | 363 |
2A The Dirichlet Problem | 366 |
2B The Heat Problem | 367 |
2C The Wave Problem | 368 |
Answers to the Exercises | 379 |
387 | |
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Expressions et termes fréquents
Algebraic Banach space bijection bounded Co(X compact operator compact sets compact subset compact support Consider contained continuous function continuous linear form converges uniformly converges weakly convex convolution Corollary countable Deduce denote dense differential Dirichlet Laplacian distribution eigenvalue element equation example exists a sequence exists a unique finite mass follows function f fundamental solution ƒ and g H¹(N hermitian operator Hilbert basis Hilbert space Hint increasing sequence inequality integer isometry L²(N Lebesgue measure Lemma Let f locally compact metric space nonempty normed space notation open set open subset orthogonal pointwise polynomial positive Radon measure Proof Proposition Prove real number relatively compact satisfies scalar product scalar product space selfadjoint sequence fn sequence n)nen Show Supp Suppose surjective Take Theorem uniform norm vector space vector subspace