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Robert Gardner Bartle (November 20, 1927 – September 18, 2003) was an American mathematician specializing in real analysis. He is known for writing the popular textbooks The Elements of Real Analysis (1964), The Elements of Integration (1966), and Introduction to Real Analysis (2011) with Donald R. Sherbert, published by John Wiley & Sons .
[3]: 30 William G. Bade and Robert G. Bartle were brought on as research assistants. [5] Dunford retired shortly after finishing the final volume. [3]: 30 Schwartz, however, went on to write similarly pathbreaking books in various other areas of mathematics. [1] [a] The book met with acclaim when published.
Rudin's text was the first modern English text on classical real analysis, and its organization of topics has been frequently imitated. [1] In Chapter 1, he constructs the real and complex numbers and outlines their properties. (In the third edition, the Dedekind cut construction is sent to an appendix for pedagogical reasons.) Chapter 2 ...
The Riemann integral uses the notion of length explicitly. Indeed, the element of calculation for the Riemann integral is the rectangle [a, b] × [c, d], whose area is calculated to be (b − a)(d − c). The quantity b − a is the length of the base of the rectangle and d − c is the height of the rectangle. Riemann could only use planar ...
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. [1] Some particular properties of real-valued sequences and functions that real analysis studies include convergence , limits , continuity , smoothness , differentiability and integrability .
First we prove the theorem for (set of all real numbers), in which case the ordering on can be put to good use. Indeed, we have the following result: Indeed, we have the following result: Lemma : Every infinite sequence ( x n ) {\displaystyle (x_{n})} in R 1 {\displaystyle \mathbb {R} ^{1}} has an infinite monotone subsequence (a subsequence ...
Bartle, Robert G.; Sherbert, Donald R. (2000). Introduction to Real Analysis (3rd ed.). New York City: John Wiley and Sons. ISBN 0-471-32148-6. Abbott, Stephen (2001). Understanding Analysis. Undergraduate Texts in Mathematics. New York: Springer Verlag. ISBN 0-387-95060-5. Rudin, Walter (1976). Principles of Mathematical Analysis. Walter Rudin ...
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear operators acting upon these spaces and respecting these structures in a suitable sense.