''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 con ...
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. Schoolbooks are textboo ...
written by
Walter Rudin
Walter may refer to:
People
* Walter (name), both a surname and a given name
* Little Walter, American blues harmonica player Marion Walter Jacobs (1930–1968)
* Gunther (wrestler), Austrian professional wrestler and trainer Walter Hahn (born 1 ...
. Initially published by
McGraw Hill
McGraw Hill is an American educational publishing company and one of the "big three" educational publishers that publishes educational content, software, and services for pre-K through postgraduate education. The company also publishes referen ...
in 1953, it is one of the most famous mathematics textbooks ever written,
[ and is renowned for its elegant and concise style of proof.
]
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 land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the ...
in the 1951–1952 academic year.W. T. Martin
William Ted Martin (June 4, 1911 – May 30, 2004) was an American mathematician, who worked on mathematical analysis, several complex variables, and probability theory. He is known for the Cameron–Martin theorem and for his 1948 book ''Several ...
, who served as a consulting editor for McGraw Hill
McGraw Hill is an American educational publishing company and one of the "big three" educational publishers that publishes educational content, software, and services for pre-K through postgraduate education. The company also publishes referen ...
, 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 thematics in a well-organized readable way, concisely, efficiently, with complete and correct proofs. It was an sthetic pleasure to work on it."
Contents
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 later editions, 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 the ...
construction is sent to an appendix for pedagogical reasons.) Chapter 2 discusses the topological properties of the real numbers as a metric space. The rest of the text covers topics such as continuous functions
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
, differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Product differentiation, in marketing
* Differentiated service, a service that varies with the identity ...
, 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 i ...
, 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 if, given any arbitra ...
), 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 ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
, 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
* Exp ...
and logarithmic functions
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
, the fundamental theorem of algebra
The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
, and Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
. After this single-variable treatment, Rudin goes in detail about real analysis in more than one dimension, with discussion of the implicit 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 ...
, and the Lebesgue integral
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Le ...
.
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