Differential Calculas in Normed Linear Spaces

1. Front Matter

   Pages i-xix

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2. Preliminaries

   • Kalyan Mukherjea

   Pages 1-36

3. Linear Spaces: New Spaces for Old

   • Kalyan Mukherjea

   Pages 37-67

4. Normed Linear Spaces, Metric Spaces

   • Kalyan Mukherjea

   Pages 68-133

5. Calculus

   • Kalyan Mukherjea

   Pages 134-208

6. Existence Theorems

   • Kalyan Mukherjea

   Pages 209-238

7. Applications: Stationary values

   • Kalyan Mukherjea

   Pages 239-267

8. Back Matter

   Pages 268-282

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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" results like the Inverse and Implicit Function Theorems but also permits, without any 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 planes and non-Euclidean geometry.

• Indian Statistical Institute, Kolkata, India

  Kalyan Mukherjea

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