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　　分類: 图书,进口原版书,科学与技术 Science & Techology ,

作者： Alberto Guzman著

出 版 社：

出版时间： 2002-7-1字数：版次： 1页数： 207印刷时间： 2002/07/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9780817642730包装： 平装内容简介

This text is an axiomatic treatment of the properties of continuous multivariable functions and related results from topology. In the context of normed 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 for 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 for the real analysis course that 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. `Continuous Functions of Vector Variables' is suitable for a course in multivariable calculus aimed at advanced undergraduates preparing for graduate programs in pure mathematics. Required background includes a course in the theory of single-variable calculus and the elements of linear algebra.

目录

Preface

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

Index

 

 

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