This balanced introduction covers all fundamentals, from the real number system and point sets to set theory and metric spaces. Useful references to the literature conclude each chapter. 1956 edition.

This concise, well-written handbook provides a distillation of real variable theory with a particular focus on the subject's significant applications to differential equations and Fourier analysis. Ample examples and brief explanations---with very few proofs and little axiomatic machinery---are used to highlight all the major results of real analysis, from the basics of sequences and series to the more advanced concepts of Taylor and Fourier series, Baire Category, and the Weierstrass Approximation Theorem. Replete with realistic, meaningful applications to differential equations, boundary value problems, and Fourier analysis, this unique work is a practical, hands-on manual of real analysis that is ideal for physicists, engineers, economists, and others who wish to use the fruits of real analysis but who do not necessarily have the time to appreciate all of the theory. Valuable as a comprehensive reference, a study guide for students, or a quick review, "A Handbook of Real Variables" will benefit a wide audience.

This undergraduate textbook is based on lectures given by the author on the differential and integral calculus of functions of several real variables. The book has a modern approach and includes topics such as: •The p-norms on vector space and their equivalence •The Weierstrass and Stone-Weierstrass approximation theorems •The differential as a linear functional; Jacobians, Hessians, and Taylor's theorem in several variables •The Implicit Function Theorem for a system of equations, proved via Banach’s Fixed Point Theorem •Applications to Ordinary Differential Equations •Line integrals and an introduction to surface integrals This book features numerous examples, detailed proofs, as well as exercises at the end of sections. Many of the exercises have detailed solutions, making the book suitable for self-study. Several Real Variables will be useful for undergraduate students in mathematics who have completed first courses in linear algebra and analysis of one real variable.

This book is written to be a convenient reference for the working scientist, student, or engineer who needs to know and use basic concepts in complex analysis. It is not a book of mathematical theory. It is instead a book of mathematical practice. All the basic ideas of complex analysis, as well as many typical applica tions, are treated. Since we are not developing theory and proofs, we have not been obliged to conform to a strict logical ordering of topics. Instead, topics have been organized for ease of reference, so that cognate topics appear in one place. Required background for reading the text is minimal: a good ground ing in (real variable) calculus will suffice. However, the reader who gets maximum utility from the book will be that reader who has had a course in complex analysis at some time in his life. This book is a handy com pendium of all basic facts about complex variable theory. But it is not a textbook, and a person would be hard put to endeavor to learn the subject by reading this book.

Designed for a first course in real variables, this text presents the fundamentals for more advanced mathematical work, particularly in the areas of complex variables, measure theory, differential equations, functional analysis, and probability. Geared toward advanced undergraduate and graduate students of mathematics, it is also appropriate for students of engineering, physics, and economics who seek an understanding of real analysis. The author encourages an intuitive approach to problem solving and offers concrete examples, diagrams, and geometric or physical interpretations of results. Detailed solutions to the problems appear within the text, making this volume ideal for independent study. Topics include metric spaces, Euclidean spaces and their basic topological properties, sequences and series of real numbers, continuous functions, differentiation, Riemann-Stieltjes integration, and uniform convergence and applications.

Real-Variable Methods in Harmonic Analysis deals with the unity of several areas in harmonic analysis, with emphasis on real-variable methods. Active areas of research in this field are discussed, from the Calderón-Zygmund theory of singular integral operators to the Muckenhoupt theory of Ap weights and the Burkholder-Gundy theory of good ? inequalities. The Calderón theory of commutators is also considered. Comprised of 17 chapters, this volume begins with an introduction to the pointwise convergence of Fourier series of functions, followed by an analysis of Cesàro summability. The discussion then turns to norm convergence; the basic working principles of harmonic analysis, centered around the Calderón-Zygmund decomposition of locally integrable functions; and fractional integration. Subsequent chapters deal with harmonic and subharmonic functions; oscillation of functions; the Muckenhoupt theory of Ap weights; and elliptic equations in divergence form. The book also explores the essentials of the Calderón-Zygmund theory of singular integral operators; the good ? inequalities of Burkholder-Gundy; the Fefferman-Stein theory of Hardy spaces of several real variables; Carleson measures; and Cauchy integrals on Lipschitz curves. The final chapter presents the solution to the Dirichlet and Neumann problems on C1-domains by means of the layer potential methods. This monograph is intended for graduate students with varied backgrounds and interests, ranging from operator theory to partial differential equations.

This is an English translation of Bourbaki’s Fonctions d'une Variable Réelle. Coverage includes: functions allowed to take values in topological vector spaces, asymptotic expansions are treated on a filtered set equipped with a comparison scale, theorems on the dependence on parameters of differential equations are directly applicable to the study of flows of vector fields on differential manifolds, etc.

This book begins with the basics of the geometry and topology of Euclidean space and continues with the main topics in the theory of functions of several real variables including limits, continuity, differentiation and integration. All topics and in particular, differentiation and integration, are treated in depth and with mathematical rigor. The classical theorems of differentiation and integration such as the Inverse and Implicit Function theorems, Lagrange's multiplier rule, Fubini's theorem, the change of variables formula, Green's, Stokes' and Gauss' theorems are proved in detail and many of them with novel proofs. The authors develop the theory in a logical sequence building one result upon the other, enriching the development with numerous explanatory remarks and historical footnotes. A number of well chosen illustrative examples and counter-examples clarify matters and teach the reader how to apply these results and solve problems in mathematics, the other sciences and economics.Each of the chapters concludes with groups of exercises and problems, many of them with detailed solutions while others with hints or final answers. More advanced topics, such as Morse's lemma, Sard's theorem , the Weierstrass approximation theorem, the Fourier transform, Vector fields on spheres, Brouwer's fixed point theorem, Whitney's embedding theorem, Picard's theorem, and Hermite polynomials are discussed in stared sections.

Based on courses given at Eötvös Loránd University (Hungary) over the past 30 years, this introductory textbook develops the central concepts of the analysis of functions of one variable — systematically, with many examples and illustrations, and in a manner that builds upon, and sharpens, the student’s mathematical intuition. The book provides a solid grounding in the basics of logic and proofs, sets, and real numbers, in preparation for a study of the main topics: limits, continuity, rational functions and transcendental functions, differentiation, and integration. Numerous applications to other areas of mathematics, and to physics, are given, thereby demonstrating the practical scope and power of the theoretical concepts treated. In the spirit of learning-by-doing, Real Analysis includes more than 500 engaging exercises for the student keen on mastering the basics of analysis. The wealth of material, and modular organization, of the book make it adaptable as a textbook for courses of various levels; the hints and solutions provided for the more challenging exercises make it ideal for independent study.