Continuous Functions of Vector Variables - Softcover

Guzman, Alberto

Softcover

ISBN 10: 0817642730 ISBN 13: 9780817642730

Publisher: Birkhäuser, 2002

Synopsis

This is an axiomatic treatment of the properties of continuous multivariable functions and related results from topology. The author covers boundedness, extreme values, and uniform continuity of functions, along with connections between continuity and topological concepts such as connectedness and compactness. The order of topics mimics the order of development in elementary calculus, with analogies and generalizations from such familiar ideas as the Pythagorean theorem.

About this title

Review

"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."

―MATHEMATICAL REVIEWS

"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."

―ZENTRALBLATT MATH

"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."

―PUBLICATIONES MATHEMATICAE, DEBRECEN

Synopsis

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, A. Guzman covers boundedness, extreme values and uniform continuity of functions, along with the connections between continuity and topological concepts such as connectedness and compactness. Suitable for advanced undergraduates preparing for graduate programmes in pure mathematics, the required background includes a course in the theory of single-variable calculus and the elements of linear algebra. 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 with 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 for the real analysis course that the book anticipates will follow this one. Applications are included where they provide interesting illustrations of the principles and theorems presented. Problems, solutions, bibliography and index complete this book.

Edition details

• Publisher: Birkhäuser
• Publication date: 2002
• Language: English
• ISBN 10: 0817642730
• ISBN 13: 9780817642730
• Binding: Paperback
• Number of pages: 210