''Principles of Mathematical Analysis'', colloquially known as ''PMA'' or ''Baby Rudin'', is an undergraduate

real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include co ...

textbook A textbook is a book containing a comprehensive compilation of content in a branch of study with the intention of explaining it. Textbooks are produced to meet the needs of educators, usually at educational institutions, but also of learners ( ...

written by

Walter Rudin Walter Rudin (May 2, 1921 – May 20, 2010) was an Austrian- American mathematician and professor of mathematics at the University of Wisconsin–Madison. In addition to his contributions to complex and harmonic analysis, Rudin was known for hi ...

. Initially published by

McGraw Hill McGraw Hill is an American education science company that provides educational content, software, and services for students and educators across various levels—from K-12 to higher education and professional settings. They produce textbooks, ...

in 1953, it is one of the most famous mathematics textbooks ever written. It is on the list of 173 books essential for undergraduate math libraries. It earned Rudin the Leroy P. Steele Prize for Mathematical Exposition in 1993.

History

As a C. L. E. Moore instructor, Rudin taught the real analysis course at

MIT The Massachusetts Institute of Technology (MIT) is a private research university in Cambridge, Massachusetts, United States. Established in 1861, MIT has played a significant role in the development of many areas of modern technology and sc ...

in the 1951–1952 academic year. After he commented to W. T. Martin, who served as a consulting editor for

, that there were no textbooks covering the course material in a satisfactory manner, Martin suggested Rudin write one himself. After completing an outline and a sample chapter, he received a contract from McGraw Hill. He completed the manuscript in the spring of 1952, and it was published the year after. Rudin noted that in writing his textbook, his purpose was "to present a beautiful area of mathematics in a well-organized readable way, concisely, efficiently, with complete and correct proofs. It was an aesthetic pleasure to work on it." The text was revised twice: first in 1964 (second edition) and then in 1976 (third edition). It has been translated into several languages, including Russian, Chinese, Spanish, French, German, Italian, Greek, Persian, Portuguese, and Polish.

Rudin's text was the first modern English text on classical real analysis, and its organization of topics has been frequently imitated. In Chapter 1, he constructs the real and complex numbers and outlines their properties. (In the third edition, the

Dedekind cut In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind (but previously considered by Joseph Bertrand), are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of a set, ...

construction is sent to an appendix for pedagogical reasons.) Chapter 2 discusses the topological properties of the real numbers as a

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

. The rest of the text covers topics such as

continuous functions In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...

, differentiation, the

Riemann–Stieltjes integral In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an inst ...

, sequences and series of functions (in particular

uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E as the function domain i ...

), and outlines examples such as

power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...

, the

exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: * Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value * Ex ...

and logarithmic functions, the

fundamental theorem of algebra The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant polynomial, constant single-variable polynomial with Complex number, complex coefficients has at least one comp ...

, and

Fourier series A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...

. After this single-variable treatment, Rudin goes in detail about real analysis in more than one dimension, with discussion of the

implicit Implicit may refer to: Mathematics * Implicit function * Implicit function theorem * Implicit curve * Implicit surface * Implicit differential equation Other uses * Implicit assumption, in logic * Implicit-association test, in social psychology * ...

and inverse function theorems,

differential forms In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...

, the

generalized Stokes theorem In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement about the integration of differential forms o ...

, and the

Lebesgue integral In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ...

References

{{Reflist

External links

Principles of Mathematical Analysis
at McGraw-Hill Education
Supplemental comments and exercises to Chapters 1-7 of Rudin
written by George Bergman Mathematical analysis Mathematics textbooks

History

Contents

References

External links