Elementary Functional Analysis

In this introductory work on mathematical analysis, the noted mathematician Georgi E. Shilov begins with an extensive and important chapter on the basic structures of mathematical analysis: linear spaces, metric spaces, normed linear spaces, Hilbert spaces, and normed algebras. The standard models for all these spaces are sets of functions (hence the term "functional analysis"), rather than sets of points in a finite-dimensional space. Chapter 2 is devoted to differential equations, and contains the basic theorems on existence and uniqueness of solutions of ordinary differential equations for functions taking values in a Banach space. The solution of the linear equation with constant (operator) coefficients is written in general form in terms of the exponential of the operator. This leads, in the finite-dimensional case, to explicit formulas not only for the solutions of first-order equations, but also to the solutions of higher-order equations and systems of equations. The third chapter presents a theory of curvature for curve in a multidimensional space. The final two chapters essentially comprise an introduction to Fourier analysis. In the treatment of orthogonal expansions, a key role is played by Fourier series and the various kinds of convergence and summability for such series. The material on Fourier transforms, in addition to presenting the more familiar theory, also deals with problems in the complex domain, in particular with problems involving the Laplace transform. Designed for students at the upper-undergraduate or graduate level, the text includes a set of problems for each chapter, with hints and answers at the end of the book. Reprint of the Prentice-Hall, 1961 edition. Preface vii 1 Basic Structures of Mathematical Analysis 1 1.1. Linear Spaces 2 1.2. Metric Spaces 25 1.3. Normed Linear Spaces 36 1.4. Hilbert Spaces 54 1.5. Approximation on a Compactum 68 1.6. Differentiation and Integration in a Normed Linear Space 83 1.7. Continuous Linear Operators 97 1.8. Normed Algebras 123 1.9. Spectral Properties of Linear Operators 131 Problems 141 2 Differential Equations 145 2.1. Definitions and Examples 145 2.2. The Fixed Point Theorem 160 2.3. Existence and Uniqueness of Solutions 163 2.4. Systems of Equations 169 2.5. Higher-Order Equations 172 2.6. Linear Equations and Systems 174 2.7. The Homogeneous Linear Equation 177 2.8. The Non-homogeneous Linear Equation 181 Problems 185 3 Space Curves 188 3.1. Basic Concepts 188 3.2. Higher Derivatives 193 3.3. Curvature 196 3.4. The Moving Basis 200 3.5. The Natural Equations 208 3.6. Helices 213 Problems 218 4 Orthogonal Expansions 220 4.1. Orthogonal Expansions in Hilbert Space 220 4.2. Trigonometric Fourier Series 226 4.3. Convergence of Fourier Series 232 4.4. Computations with Fourier Series 241 4.5. Divergent Fourier Series and Generalized Summation 258 4.6. Other Orthogonal Systems 264 Problems 270 5 The Fourier Transform 274 5.1. The Fourier Integral and Its Inversion 274 5.2. Further Properties of the Fourier Transform 280 5.3. Examples and Applications 293 5.4. The Laplace Transform 296 5.5. Quasi-Analytic Classes of Functions 307 Problems 317 Hints and Answers 320 Bibliography 326 Index 327 Introductory text covers basic structures of mathematical analysis (linear spaces, metric spaces, normed linear spaces, etc.), differential equations, orthogonal expansions, Fourier transforms — including problems in the complex domain, especially involving the Laplace transform — and more. Each chapter includes a set of problems, with hints and answers. Bibliography. 1974 edition.