Real Analysis Neal Carothers

Advanced Real Analysis systematically develops those concepts and tools in real analysis that are vital to every mathematician, whether pure or applied, aspiring or established. Along with a companion volume Basic Real Analysis (available separately or together as a Set), these works present a comprehensive treatment with a global view of the subject, emphasizing the connections between real analysis and other branches of mathematics.

Advanced Real Analysis systematically develops those concepts and tools in real analysis that are vital to every mathematician, whether pure or applied, aspiring or established. Along with a companion volume Basic Real Analysis (available separately or together as a Set), these works present a comprehensive treatment with a global view of the subject, emphasizing the connections between real analysis and other branches of mathematics.

This text is a rigorous, detailed introduction to real analysis that presents the fundamentals with clear exposition and carefully written definitions, theorems, and proofs. The choice of material and the flexible organization, including three different entryways into the study of the real numbers, making it equally appropriate to undergraduate mathematics majors who want to continue in mathematics, and to future mathematics teachers who want to understand the theory behind calculus.

This new approach to real analysis stresses the use of the subject in applications, showing how the principles and theory of real analysis can be applied in various settings. Applications cover approximation by polynomials, discrete dynamical systems, differential equations, Fourier series and physics, Fourier series and approximation, wavelets, and convexity and optimization. Each chapter has many useful exercises.

Mathematics is the music of science, and real analysis is the Bach of mathematics. There are many other foolish things I could say about the subject of this book, but the foregoing will give the reader an idea of where my heart lies. The present book was written to support a first course in real analysis, normally taken after a year of elementary calculus. Real analysis is, roughly speaking, the modern setting for Calculus, "real" alluding to the field of real numbers that underlies it all. At center stage are functions, defined and taking values in sets of real numbers or in sets (the plane, 3-space, etc.) readily derived from the real numbers; a first course in real analysis traditionally places the emphasis on real-valued functions defined on sets of real numbers.

This textbook (OER more formally) is a course in undergraduate real analysis (somewhere it is called "advanced calculus"). The book is meant both for a basic course for students who do not necessarily wish to go to graduate school, but also as a more advanced course that also covers topics such as metric spaces and should prepare students for graduate study. A prerequisite for the course is a basic proof course. Sample Darboux sumsAn advanced course could be two semesters long with some of the second-semester topics such as multivariable differential calculus, path integrals, and the multivariable integral using the second volume. There are more topics than can be covered in two semesters, and it can also be reading for beginning graduate students to refresh their analysis or fill in some of the holes.

The book targets undergraduate and postgraduate mathematics students and helps them develop a deep understanding of mathematical analysis. Designed as a first course in real analysis, it helps students learn how abstract mathematical analysis solves mathematical problems that relate to the real world. As well as providing a valuable source of inspiration for contemporary research in mathematics, the book helps students read, understand and construct mathematical proofs, develop their problem-solving abilities and comprehend the importance and frontiers of computer facilities and much more.

From the point of view of strict logic, a rigorous course on real analysis should precede a course on calculus. Strict logic, is, however, overruled by both history and practicality. Historically, calculus, with its origins in the 17th century, came first, and made rapid progress on the basis of informal intuition.