Postmodern Analysis

Authors:

ISBN: 978-3-540-43873-1 (Print) 978-3-662-05306-5 (Online)

Table of contents (26 chapters)

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  1. Front Matter

    Pages I-XVII

  2. Calculus for Functions of One Variable

    1. Front Matter

      Pages 1-1

    2. No Access

      Book Chapter

      Pages 3-12

      Prerequisites

    3. No Access

      Book Chapter

      Pages 13-19

      Limits and Continuity of Functions

    4. No Access

      Book Chapter

      Pages 21-29

      Differentiability

    5. No Access

      Book Chapter

      Pages 31-42

      Characteristic Properties of Differentiable Functions. Differential Equations

    6. No Access

      Book Chapter

      Pages 43-46

      The Banach Fixed Point Theorem. The Concept of Banach Space

    7. No Access

      Book Chapter

      Pages 47-60

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

    8. No Access

      Book Chapter

      Pages 61-73

      Integrals and Ordinary Differential Equations

  3. Topological Concepts

    1. Front Matter

      Pages 75-75

    2. No Access

      Book Chapter

      Pages 77-99

      Metric Spaces: Continuity, Topological Notions, Compact Sets

  4. Calculus in Euclidean and Banach Spaces

    1. Front Matter

      Pages 101-101

    2. No Access

      Book Chapter

      Pages 103-114

      Differentiation in Banach Spaces

    3. No Access

      Book Chapter

      Pages 115-131

      Differential Calculus in ℝd

    4. No Access

      Book Chapter

      Pages 133-143

      The Implicit Function Theorem. Applications

    5. No Access

      Book Chapter

      Pages 145-153

      Curves in ℝ d . Systems of ODEs

  5. The Lebesgue Integral

    1. Front Matter

      Pages 155-155

    2. No Access

      Book Chapter

      Pages 157-163

      Preparations. Semicontinuous Functions

    3. No Access

      Book Chapter

      Pages 165-182

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

    4. No Access

      Book Chapter

      Pages 183-193

      Lebesgue Integrable Functions and Sets

    5. No Access

      Book Chapter

      Pages 195-203

      Null Functions and Null Sets. The Theorem of Fubini

    6. No Access

      Book Chapter

      Pages 205-215

      The Convergence Theorems of Lebesgue Integration Theory

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      Book Chapter

      Pages 217-227

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

    8. No Access

      Book Chapter

      Pages 229-237

      The Transformation Formula

  6. L p and Sobolev Spaces

    1. Front Matter

      Pages 239-239

    2. No Access

      Book Chapter

      Pages 241-260

      The L p -Spaces

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