Serov V. Classical Analysis of Real-Valued Functions 2023

Textbook in PDF format

Divided into two self-contained parts, this textbook is an introduction to modern real analysis. More than 350 exercises and 100 examples are integrated into the text to help clarify the theoretical considerations and the practical applications to differential geometry, Fourier series, differential equations, and other subjects. The first part of Classical Analysis of Real-Valued Functions covers the theorems of existence of supremum and infimum of bounded sets on the real line and the Lagrange formula for differentiable functions. Applications of these results are crucial for classical mathematical analysis, and many are threaded through the text. In the second part of the book, the implicit function theorem plays a central role, while the Gauss–Ostrogradskii formula, surface integration, Heine–Borel lemma, the Ascoli–Arzelà theorem, and the one-dimensional indefinite Lebesgue integral are also covered. This book is intended for first and second year students majoring in mathematics although students of engineering disciplines will also gain important and helpful insights. It is appropriate for courses in mathematical analysis, functional analysis, real analysis, and calculus and can be used for self-study as well.
List of Figures.
Preface.
Introduction.
Real numbers.
Theory of limits of real numbers.
Series and infinite products of real numbers.
Continuity of one-variable functions.
Differentiation.
Taylor’s expansion and its applications.
Indefinite Riemann integral.
Riemann–Stieltjes integral.
Improper integrals.
Approximate methods.
Geometric applications of integrals. Elementary measure theory.
Continuity and differentiability of functions of several variables.
Implicit functions.
Multidimensional Riemann integrals.
Lebesgue integration.
Continuity in Banach spaces. Topological concepts.
Bibliography.
Index