It also remains the single most assigned text for undergraduate real analysis by professors. Moore Instructor at MIT in the early s. Rudin was discussing the difficulty of choosing a suitable text with Ted Martin, then chair of the mathematics department at MIT. Back then, there simply were no modern texts on classical real analysis in English. More importantly, they were too advanced for such a course. Martin quite naturally suggested Rudin write such a text.

Chapter 1 gives a detailed study of the real number field. While it may be more indirect, this method is overall a little easier for students to digest, particularly those with some training in set theory or algebra.

## Principles of Mathematical Analysis

Chapter 2, taken with its exercises, gives a very complete account of the topological properties of R as a metric space. Chapter 3, on numerical sequences and series, is, to me, the best chapter in the book.

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This chapter also pretty much sets the tone for the rest of the book: It is crystal clear but concise to a brutal degree. Proofs and most examples are stated brusquely with little or no further explanation beyond the definition and the statement of fact. Chapter 4 discusses continuous functions in metric spaces, the relation between connectedness and continuity in metric spaces, compactness in such spaces, discontinuities, infinite limits and uniform continuity. Chapter 7 covers sequences and series of functions and their convergence properties, focusing on uniform convergence as the central notion for developing the properties of function spaces of real valued maps, such as equicontinuity and the Stone-Weierstrass theorem.

Chapter 8 uses these properties to develop some useful functions of real analysis, such as power series, Fourier series and the Gamma function. The result is more confusing than informative. The book concludes with a quick overview of the Lebesgue integral in chapter 11, which seems tacked on and forced. Prior to chapter 9, however, the book gives a masterly presentation of real analysis for the serious math student. Despite its conciseness, Rudin does for the most part supply enough detail that able students can fill in the blanks.

One does wish Rudin had included more examples in the presentation. Surprisingly, Rudin usually does give excellent hints pointing the student in the right direction. The Basic Library List Committee considers this book essential for undergraduate mathematics libraries. Steven G. At MIT, the book has been practically canonized: I was once visited by some of my friends taking math in Cambridge and I was angrily dismissed as an ignorant dabbler for even suggesting any other text for undergraduate real analysis even existed. Love it or hate it, the book elicits incredibly strong passions in people.

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It also remains the single most assigned text for undergraduate real analysis by professors. Moore Instructor at MIT in the early s.

Rudin was discussing the difficulty of choosing a suitable text with Ted Martin, then chair of the mathematics department at MIT. Back then, there simply were no modern texts on classical real analysis in English. More importantly, they were too advanced for such a course.

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Martin quite naturally suggested Rudin write such a text. Chapter 1 gives a detailed study of the real number field. While it may be more indirect, this method is overall a little easier for students to digest, particularly those with some training in set theory or algebra.

Chapter 2, taken with its exercises, gives a very complete account of the topological properties of R as a metric space. Chapter 3, on numerical sequences and series, is, to me, the best chapter in the book. This chapter also pretty much sets the tone for the rest of the book: It is crystal clear but concise to a brutal degree. Proofs and most examples are stated brusquely with little or no further explanation beyond the definition and the statement of fact.

## Concise Introduction to Basic Real Analysis - CRC Press Book

Chapter 4 discusses continuous functions in metric spaces, the relation between connectedness and continuity in metric spaces, compactness in such spaces, discontinuities, infinite limits and uniform continuity. Chapter 7 covers sequences and series of functions and their convergence properties, focusing on uniform convergence as the central notion for developing the properties of function spaces of real valued maps, such as equicontinuity and the Stone-Weierstrass theorem.

Chapter 8 uses these properties to develop some useful functions of real analysis, such as power series, Fourier series and the Gamma function. The result is more confusing than informative. The book concludes with a quick overview of the Lebesgue integral in chapter 11, which seems tacked on and forced.