Differential Calculus in Normed Linear Spaces/Kalyan Mukherjea. Second Edition. New Delhi, Hindustan Book Agency, 2007, xx, 282 p., figs., (pbk). ISBN 81-85931-76-3. [Texts and Readings in Mathematics-26].

    Contents: Preface. How to use this book. 1. Preliminaries. 2. Linear spaces: new spaces for old. 3. Normed linear spaces, metric spaces. 4. Calculus. 5. Existence theorems. 6. Applications: stationary values. Appendix: Green's theorem. References. Index.

    "This book presents Advanced Calculus from a geometric point of view: instead of dealing with partial derivatives of functions of several variables, the derivative of the function is treated as a linear transformation between normed linear spaces. Not only does this lead to a simplified and transparent exposition of "difficult" result like the inverse and implicit function theorems but also permits, without only extra effort, a discussion of the differential calculus of functions defined on infinite dimensional Hilbert or Banach spaces.

    The prerequisites demanded of the reader are modest: a sound understanding of convergence of sequences and series of real numbers, the continuity and differentiability properties of functions of a real variable and a little linear Algebra should provide adequate background for understanding the book. The first two chapters cover much of the more advanced background material on Linear Algebra (like dual spaces, multilinear functions and tensor products). Chapter 3 gives an ab initio exposition of the basic results concerning the topology of metric spaces, particularly of normed linear spaces.

    The last chapter deals with miscellaneous applications of the differential calculus including an introduction to the calculus of variations. As a corollary to this, there is a brief discussion of geodesics in Euclidean and hyperbolic planers and non-Euclidean geometry."

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