Postmoderne Analysis

"This is an introduction to advanced analysis at the beginning graduate level that blends a modern presentation with concrete examples and applications, in particular in the areas of calculus of variations and partial differential equations. The book does not strive for abstraction for its own sake, but tries rather to impart a working knowledge of the key methods of contemporary analysis, in particular those that are also relevant for application in physics. It provides a streamlined and quick introduction to the fundamental concepts of Banach space and Lebesgue integration theory and the basic notions of the calculus of variations, including Sobolev space theory." "The new edition contains additional material on the qualitative behavior of solutions of ordinary differential equations, some further details on L[superscript p] and Sobolev functions, partitions of unity and a brief introduction to abstract measure theory."--Jacket Front Matter....Pages I-XVII Front Matter....Pages 1-1 Prerequisites....Pages 3-12 Limits and Continuity of Functions....Pages 13-19 Differentiability....Pages 21-29 Characteristic Properties of Differentiable Functions. Differential Equations....Pages 31-42 The Banach Fixed Point Theorem. The Concept of Banach Space....Pages 43-46 Uniform Convergence. Interchangeability of Limiting Processes. Examples of Banach Spaces. The Theorem of Arzela-Ascoli....Pages 47-60 Integrals and Ordinary Differential Equations....Pages 61-73 Front Matter....Pages 75-75 Metric Spaces: Continuity, Topological Notions, Compact Sets....Pages 77-99 Front Matter....Pages 101-101 Differentiation in Banach Spaces....Pages 103-114 Differential Calculus in R d ....Pages 115-131 The Implicit Function Theorem. Applications....Pages 133-143 Curves in R d . Systems of ODEs....Pages 145-153 Front Matter....Pages 155-155 Preparations. Semicontinuous Functions....Pages 157-163 The Lebesgue Integral for Semicontinuous Functions. The Volume of Compact Sets....Pages 165-182 Lebesgue Integrable Functions and Sets....Pages 183-193 Null Functions and Null Sets. The Theorem of Fubini....Pages 195-203 The Convergence Theorems of Lebesgue Integration Theory....Pages 205-215 Measurable Functions and Sets. Jensen’s Inequality. The Theorem of Egorov....Pages 217-227 The Transformation Formula....Pages 229-237 Front Matter....Pages 239-239 The L p -Spaces....Pages 241-260 Front Matter....Pages 239-239 Integration by Parts. Weak Derivatives. Sobolev Spaces....Pages 261-282 Front Matter....Pages 283-283 Hilbert Spaces. Weak Convergence....Pages 285-294 Variational Principles and Partial Differential Equations....Pages 295-326 Regularity of Weak Solutions....Pages 327-341 The Maximum Principle....Pages 343-353 The Eigenvalue Problem for the Laplace Operator....Pages 355-360 Back Matter....Pages 361-371 What is the title of this book intended to signify, what connotations is the adjective "Postmodern" meant to carry? A potential reader will surely pose this question. To answer it, I should describe what distinguishes the approach to analysis presented here from what has been called "Modern Analysis" by its protagonists. "Modern Analysis" as represented in the works of the Bour­ baki group or in the textbooks by Jean Dieudonne is characterized by its systematic and axiomatic treatment and by its drive towards a high level of abstraction. Given the tendency of many prior treatises on analysis to degen­ erate into a collection of rather unconnected tricks to solve special problems, this definitely represented a healthy achievement. In any case, for the de­ velopment of a consistent and powerful mathematical theory, it seems to be necessary to concentrate solely on the internal problems and structures and to neglect the relations to other fields of scientific, even of mathematical study for a certain while. Almost complete isolation may be required to reach the level of intellectual elegance and perfection that only a good mathematical theory can acquire. However, once this level has been reached, it might be useful to open one's eyes again to the inspiration coming from concrete ex­ ternal problems.