Continuous Functions of Vector Variables

Häftad, Engelska, 2002

699 kr

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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, 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.

Produktinformation

Utgivningsdatum2002-07-31
Mått155 x 235 x 13 mm
Vikt341 g
FormatHäftad
SpråkEngelska
Antal sidor210
Upplaga2002
FörlagBirkhauser Boston
ISBN9780817642730

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Beräkning och matematisk analys inom Naturvetenskap och teknik

1 Euclidean Space.- 1.1 Multiple Variables.- 1.2 Points and Lines in a Vector Space.- 1.3 Inner Products and the Geometry of Rn.- 1.4 Norms and the Definition of Euclidean Space.- 1.5 Metrics.- 1.6 Infinite-Dimensional Spaces.- 2 Sequences in Normed Spaces.- 2.1 Neighborhoods in a Normed Space.- 2.2 Sequences and Convergence.- 2.3 Convergence in Euclidean Space.- 2.4 Convergence in an Infinite-Dimensional Space.- 3 Limits and Continuity in Normed Spaces.- 3.1 Vector-Valued Functions in Euclidean Space.- 3.2 Limits of Functions in Normed Spaces.- 3.3 Finite Limits.- 3.4 Continuity.- 3.5 Continuity in Infinite-Dimensional Spaces.- 4 Characteristics of Continuous Functions.- 4.1 Continuous Functions on Boxes in Euclidean Space.- 4.2 Continuous Functions on Bounded Closed Subsets of Euclidean Space.- 4.3 Extreme Values and Sequentially Compact Sets.- 4.4 Continuous Functions and Open Sets.- 4.5 Continuous Functions on Connected Sets.- 4.6 Finite-Dimensional Subspaces of Normed Linear Spaces.- 5 Topology in Normed Spaces.- 5.1 Connected Sets.- 5.2 Open Sets.- 5.3 Closed Sets.- 5.4 Interior, Boundary, and Closure.- 5.5 Compact Sets.- 5.6 Compactness in Infinite Dimensions.- Solutions to Exercises.- References.

"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