Continuous Functions of Vector Variables / Edition 1

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This text is an axiomatic treatment of the properties of continuous multivariable functions and related results from topology. Viewing multiple real variables as members of vector spaces, the author covers boundedness, extreme values, and uniform continuity of functions, along with the connections between continuity and topological concepts such as connectedness and compactness. The order of topics deliberately mimics the order of development in elementary calculus. This sequencing allows an elementary approach, with analogies to and generalizations from such familiar ideas as the Pythagorean Theorem. The reader is frequently reminded that the pictures suggested by geometry are powerful guides and tools. The definition-theorem-proof format resides within an informal exposition, containing numerous historical comments and questions within and between the proofs. The objective is to present precise proofs, but in a structure and tone that teach the student to plan and write proofs, both in general and specifically in a real analysis course. Applications are largely absent, except where they provide interesting illustrations of the principles and theorems presented. Problems, solutions, bibliography and index complete this book. Continuous Functions of Vector Variables is suitable for advanced undergraduates preparing for graduate programs in pure mathematics. Required background includes a course in the theory of one-variable calculus and the elements of linear algebra.

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Editorial Reviews

From the Publisher
"The book is devoted to mathematical analysis in multi-dimensional spaces, including infinite-dimensional ones…One of the important characteristics of the text is the topological approach. …It is worth mentioning that all these ideas are presented in conformity with the situation, i.e. they are strictly related to normed spaces and no unnecessary abstract setting is used. … The language is very clear and sometimes very ‘fresh’ if compared with classical textbooks, so that it may better appeal to the reader. There are many exercises, and the book concludes with a chapter containing solutions."


"This is a textbook on calculus of several variables. It covers algebraic and metric structure of the Euclidean space, convergence, basic properties of continuous functions and topology in normed spaces.... In my opinion, this topological approach is one of the advantages of the book under review. It is written very clearly and contains numerous examples and instructive pictures. Each section is endowed with a set of exercises, and the book is concluded with solutions to these exercises. This is a nice and user-friendly textbook which can be recommended for a one-semester course in multivariable calculus."


"The presentation is detailed and clear. The leisurely discursive style adopted by the author will be appreciated by beginners. The fact that the solution of each and every exercise can be found at the end of the book, considerable enhances its value and makes it suitable also for individual study. This book successfully bridges the gap between elemntary calculus and such higher disciplines as real functions, general topology, and, in particular, functional analysis."


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Product Details

  • ISBN-13: 9780817642730
  • Publisher: Birkhauser Verlag
  • Publication date: 7/31/2002
  • Edition description: 2002
  • Edition number: 1
  • Pages: 210
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.46 (d)

Table of Contents

• Euclidian Space
• Sequences In Normed Spaces
• Limits and Continuity in Normed Spaces
• Characteristics of Continuous Functions
• Topology in Normed Spaces
• Answers
• Bibliography
• Index

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