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Preface

1. Basic Structures of Mathematical Analysis

1.1 Linear Spaces

1.2 Metric Spaces

1.3 Normed Linear Spaces

1.4 Hilbert Spaces

1.5 Approximation on a Compactum

1.6 Differentiation and Integration in a Normed Linear Space

1.7 Continuous Linear Operators

1.8 Normed Algebras

1.9 Spectral Properties of Linear Operators

Problems

2. Differential Equations

2.1 Definitions and Examples

2.2 The Fixed Point Theorem

2.3 Existence and Uniqueness of Solutions

2.4 Systems of Equations

2.5 Higher-Order Equations

2.6 Linear Equations and systems

2.7 The Homogeneous Linear Equation

2.8 The Nonhomogeneous Linear Equation

Problems

3. Space Curves

3.1 Basic Concepts

3.2 Higher Derivatives

3.3 Curvature

3.4 The Moving Basis

3.5 The Natural Equations

3.6 Helices

Problems

4. Orthogonal Expansions

4.1 Orthogonal Expansions in Hilbert Space

4.2 Trigonometric Fourier Series

4.3 Convergence of Fourier Series

4.4 Computations with Fourier Series

4.5 Divergent Fourier Series and Generalized Summation

4.6 Other Orthogonal Systems

Problems

5. The Fourier Transform

5.1 The Fourier Integral and Its Inversion

5.2 Further Properties of the Fourier Transform

5.3 Examples and Applications

5.4 The Laplace Transform

5.5 Quasi-Analytic Classes of Functions

Problems

Hints and Answers Bibliography Index

1. Basic Structures of Mathematical Analysis

1.1 Linear Spaces

1.2 Metric Spaces

1.3 Normed Linear Spaces

1.4 Hilbert Spaces

1.5 Approximation on a Compactum

1.6 Differentiation and Integration in a Normed Linear Space

1.7 Continuous Linear Operators

1.8 Normed Algebras

1.9 Spectral Properties of Linear Operators

Problems

2. Differential Equations

2.1 Definitions and Examples

2.2 The Fixed Point Theorem

2.3 Existence and Uniqueness of Solutions

2.4 Systems of Equations

2.5 Higher-Order Equations

2.6 Linear Equations and systems

2.7 The Homogeneous Linear Equation

2.8 The Nonhomogeneous Linear Equation

Problems

3. Space Curves

3.1 Basic Concepts

3.2 Higher Derivatives

3.3 Curvature

3.4 The Moving Basis

3.5 The Natural Equations

3.6 Helices

Problems

4. Orthogonal Expansions

4.1 Orthogonal Expansions in Hilbert Space

4.2 Trigonometric Fourier Series

4.3 Convergence of Fourier Series

4.4 Computations with Fourier Series

4.5 Divergent Fourier Series and Generalized Summation

4.6 Other Orthogonal Systems

Problems

5. The Fourier Transform

5.1 The Fourier Integral and Its Inversion

5.2 Further Properties of the Fourier Transform

5.3 Examples and Applications

5.4 The Laplace Transform

5.5 Quasi-Analytic Classes of Functions

Problems

Hints and Answers Bibliography Index

Library of Congress subject headings for this publication: Functional analysis