TY  - BOOK
AU  - Taylor, Joseph L.
TI  - Foundations of analysis
T2  - Pure and applied undergraduate texts
SN  - 9781470425791
U1  - 517 
PY  - 2016///
CY  - Hyderabad
PB  - Universities Press
KW  - Functional analysis
KW  - Functions of real variables
KW  - Real functions -- Instructional exposition (textbooks, tutorial papers, etc.)
KW  - Real functions -- Functions of one variable -- Functions of one variable.
KW  - Real functions -- Functions of several variables -- Functions of several variables
KW  - Real functions -- Inequalities -- Inequalities
KW  - Mathematical logic and foundations -- Set theory -- Set theory
N1  - Includes bibliographical references (page 387) and index
N2  - Analysis plays a crucial role in the undergraduate curriculum. Building upon the familiar notions of calculus, analysis introduces the depth and rigor characteristic of higher mathematics courses. Foundations of Analysis has two main goals. The first is to develop in students the mathematical maturity and sophistication they will need as they move through the upper division curriculum. The second is to present a rigorous development of both single and several variable calculus, beginning with a study of the properties of the real number system.

The presentation is both thorough and concise, with simple, straightforward explanations. The exercises differ widely in level of abstraction and level of difficulty. They vary from the simple to the quite difficult and from the computational to the theoretical. Each section contains a number of examples designed to illustrate the material in the section and to teach students how to approach the exercises for that section.

The list of topics covered is rather standard, although the treatment of some of them is not. The several variable material makes full use of the power of linear algebra, particularly in the treatment of the differential of a function as the best affine approximation to the function at a given point. The text includes a review of several linear algebra topics in preparation for this material. In the final chapter, vector calculus is presented from a modern point of view, using differential forms to give a unified treatment of the major theorems relating derivatives and integrals: Green's, Gauss's, and Stokes's Theorems.

At appropriate points, abstract metric spaces, topological spaces, inner product spaces, and normed linear spaces are introduced, but only as asides. That is, the course is grounded in the concrete world of Euclidean space, but the students are made aware that there are more exotic worlds in which the concepts they are learning may be studied
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