Postmodern Analysis

1. Front Matter

   Pages I-XV

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2. Calculus for Functions of One Variable

   1. Prerequisites

      Pages 3-12

   2. Limits and Continuity of Functions

      Pages 13-19

   3. Differentiability

      Pages 21-29

   4. Characteristic Properties of Differentiable Functions. Differential Equations

      Pages 31-42

   5. The Banach Fixed Point Theorem. The Concept of Banach Space

      Pages 43-46

   6. Uniform Convergence. Interchangeability of Limiting Processes. Examples of Banach Spaces. The Theorem of Arzela-Ascoli

      Pages 47-60

   7. Integrals and Ordinary Differential Equations

      Pages 61-73

3. Topological Concepts

   1. Metric Spaces: Continuity, Topological Notions, Compact Sets

      Pages 77-99

4. Calculus in Euclidean and Banach Spaces

   1. Differentiation in Banach Spaces

      Pages 103-114

   2. Differential Calculus in \(\mathbb{R}\)^d

      Pages 115-131

   3. The Implicit Function Theorem. Applications

      Pages 133-143

   4. Curves in \(\mathbb{R}\)^d. Systems of ODEs

      Pages 145-153

5. The Lebesgue Integral

   1. Preparations. Semicontinuous Functions

      Pages 157-163

   2. The Lebesgue Integral for Semicontinuous Functions. The Volume of Compact Sets

      Pages 165-182

   3. Lebesgue Integrable Functions and Sets

      Pages 183-193

   4. Null Functions and Null Sets. The Theorem of Fubini

      Pages 195-203

   5. The Convergence Theorems of Lebesgue Integration Theory

      Pages 205-215

   6. Measurable Functions and Sets. Jensen’s Inequality. The Theorem of Egorov

      Pages 217-227

   7. The Transformation Formula

      Pages 229-237

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 - proach to analysis presented here from what has by its protagonists been called “Modern Analysis”. “Modern Analysis” as represented in the works of the Bourbaki group or in the textbooks by Jean Dieudonn´ e 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 degenerate into a collection of rather unconnected tricks to solve special problems, this de?nitely represented a healthy achievement. In any case, for the development 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 ?elds of scienti?c, 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 mathem- ical theory can acquire. However, once this level has been reached, it can be useful to open one’s eyes again to the inspiration coming from concrete external problems.

• Banach and Hilbert space
• Continuous and differentiable functions
• Lebesgue integration theory
• Sobolev space
• Sobolov spaces
• analysis
• calculus of variations
• differential equations
• implicit function theorem
• partial differential equations

• Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

  Jürgen Jost

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