This textbook explores the foundations of real analysis using the framework of general ordered fields, demonstrating the multifaceted nature of the area. Focusing on the logical structure of real analysis, the definitions and interrelations between core concepts are illustrated with the use of numerous examples and counterexamples. Readers will learn of the equivalence between various theorems and the completeness property of the underlying ordered field. These equivalences emphasize the fundamental role of real numbers in analysis. Comprising six chapters, the book opens with a rigorous presentation of the theories of rational and real numbers in the framework of ordered fields. This is followed by an accessible exploration of standard topics of elementary real analysis, including continuous functions, differentiation, integration, and infinite series. Readers will find this text conveniently self-contained, with three appendices included after the main text, covering an overview of natural numbers and integers, Dedekind's construction of real numbers, historical notes, and selected topics in algebra. Real Analysis: Foundations is ideal for students at the upper-undergraduate or beginning graduate level who are interested in the logical underpinnings of real analysis. With over 130 exercises, it is suitable for a one-semester course on elementary real analysis, as well as independent study.

Definitive look at modern analysis, with views of applications to statistics, numerical analysis, Fourier series, differential equations, mathematical analysis, and functional analysis. More than 750 exercises; some hints and solutions. 1981 edition.

A Readable yet Rigorous Approach to an Essential Part of Mathematical Thinking Back by popular demand, Real Analysis and Foundations, Third Edition bridges the gap between classic theoretical texts and less rigorous ones, providing a smooth transition from logic and proofs to real analysis. Along with the basic material, the text covers Riemann-Stieltjes integrals, Fourier analysis, metric spaces and applications, and differential equations. New to the Third Edition Offering a more streamlined presentation, this edition moves elementary number systems and set theory and logic to appendices and removes the material on wavelet theory, measure theory, differential forms, and the method of characteristics. It also adds a chapter on normed linear spaces and includes more examples and varying levels of exercises. Extensive Examples and Thorough Explanations Cultivate an In-Depth Understanding This best-selling book continues to give students a solid foundation in mathematical analysis and its applications. It prepares them for further exploration of measure theory, functional analysis, harmonic analysis, and beyond.

Foundations of Analysis has two main goals. The first is to develop in students the mathematical maturity and sophistication they will need as they move through the upper division curriculum. The second is to present a rigorous development of both single and several variable calculus, beginning with a study of the properties of the real number system. The presentation is both thorough and concise, with simple, straightforward explanations. The exercises differ widely in level of abstraction and level of difficulty. They vary from the simple to the quite difficult and from the computational to the theoretical. Each section contains a number of examples designed to illustrate the material in the section and to teach students how to approach the exercises for that section. --Book cover.

A complete course on metric, normed, and Hilbert spaces, including many results and exercises seldom found in texts on analysis at this level. The author covers an unusually wide range of material in a clear and concise format, including elementary real analysis, Lebesgue integration on R, and an introduction to functional analysis. The book begins with a fast-paced course on real analysis, followed by an introduction to the Lebesgue integral. This provides a reference for later chapters as well as a preparation for students with only the typical sequence of undergraduate calculus courses as prerequisites. Other features include a chapter introducing functional analysis, the Hahn-Banach theorem and duality, separation theorems, the Baire Category Theorem, the Open Mapping Theorem and their consequences, and unusual applications. Of special interest are the 750 exercises, many with guidelines for their solutions, applications and extensions of the main propositions and theorems, pointers to new branches of the subject, and difficult challenges for the very best students.

This textbook covers the subject of real analysis from the fundamentals up through beginning graduate level. It is appropriate as an introductory course text or a review text for graduate qualifying examinations. Some special features of the text include a thorough discussion of transcendental functions such as trigonometric, logarithmic, and exponential from power series expansions, deducing all important functional properties from the series definitions. The text is written in a user-friendly manner, and includes full solutions to all assigned exercises throughout the text.

This book is intended as a serious introduction to the studyof mathematical analysis. In contrast to calculus, mathematical analysis does not involve formula manipulation, memorizing integrals or applications to other fields of science. No.It involves geometric intuition and proofs of theorems. It ispure mathematics! Given the mathematical preparation andinterest of our intended audience which, apart from mathematics majors, includes students of statistics, computer science, physics, students of mathematics education and students of engineering, we have not given the axiomatic development of the real number system. However, we assumethat the reader is familiar with sets and functions. This bookis divided into two parts. Part I covers elements of mathematical analysis which include: the real number system, bounded subsets of real numbers, sequences of real numbers, monotone sequences, Bolzano-Weierstrass theorem, Cauchysequences and completeness of R, continuity, intermediatevalue theorem, continuous maps on [a, b], uniform continuity, closed sets, compact sets, differentiability, series of nonnegative real numbers, alternating series, absolute and conditional convergence; and re-arrangement of series. The contents of Part I are adequate for a semester course in mathematical analysis at the 200 level. Part II covers Riemannintegrals. In particular, the Riemann integral, basic properties of Riemann integral, pointwise convergence of sequencesof functions, uniform convergence of sequences of functions, series of real-valued functions: term by term differentiationand integration; power series: uniform convergence of powerseries; uniform convergence at end points; and equi-continuity are covered. Part II covers the standard syllabus for asemester mathematical analysis course at the 300 level. Thetopics covered in this book provide a reasonable preparationfor any serious study of higher mathematics. But for one toreally benefit from the book, one must spend a great deal ofixtime on it, studying the contents very carefully and attempting all the exercises, especially the miscellaneous exercises atthe end of the book. These exercises constitute an importantintegral part of the book.Each chapter begins with clear statements of the most important theorems of the chapter. The proofs of these theoremsgenerally contain fundamental ideas of mathematical analysis. Students are therefore encouraged to study them verycarefully and to discover these id