An entire generation of mathematicians has grown up during the time - tween the appearance of the ?rst edition of this textbook and the publication of the fourth edition, a translation of which is before you. The book is fam- iar to many people, who either attended the lectures on which it is based or studied out of it, and who now teach others in universities all over the world. I am glad that it has become accessible to English-speaking readers. This textbook consists of two parts. It is aimed primarily at university students and teachers specializing in mathematics and natural sciences, and at all those who wish to see both the rigorous mathematical theory and examplesofitse?ectiveuseinthesolutionofrealproblemsofnaturalscience. The textbook exposes classical analysis as it is today, as an integral part of Mathematics in its interrelations with other modern mathematical courses such as algebra, di?erential geometry, di?erential equations, complex and functional analysis.
The thoroughness of the coverage, from elementary to very advanced
The clarity of the exposition
The originality and variety of the exercises and examples
Especially good for physics studentsInhalt
CONTENTS OF VOLUME II Prefaces Preface to the fourth edition Prefact to the third edition Preface to the second edition Preface to the first edition 9* Continuous Mappings (General Theory) 9.1 Metric spaces 9.1.1 Definitions and examples 9.1.2 Open and closed subsets of a metric space 9.1.3 Subspaces of a metric space 9.1.4 The direct product of metric spaces 9.1.5 Problems and exercises 9.2 Topological spaces 9.2.1 Basic definitions 9.2.2 Subspaces of a topological space 9.2.3 The direct product of topological spaces 9.2.4 Problems and exercises 9.3 Compact sets 9.3.1 Definition and general properties of compact sets 9.3.2 Metric compact sets 9.3.3 Problems and exercises 9.4 Connected topological spaces 9.4.1 Problems and exercises 9.5 Complete metric spaces 9.5.1 Basic definitions and examples 9.5.2 The completion of a metric space 9.5.3 Problems and exercises 9.6 Continuous mappings of topological spaces 9.6.1 The limit of a mapping 9.6.2 Continuous mappings 9.6.3 Problems and exercises 9.7 The contraction mapping principle 9.7.1 Problems and exercises 10 *Differential Calculus from a General Viewpoint 10.1 Normed vector spaces 10.1.1 Some examples of the vector spaces of analysis 10.1.2 Norms in vector spaces 10.1.3 Inner products in a vector space 10.1.4 Problems and exercises 10.2 Linear and multilinear transformations 10.2.1 Definitions and examples 10.2.2 The norm of a transformation 10.2.3 The space of continuous transformations 10.2.4 Problems and exercises 10.3 The differential of a mapping 10.3.1 Mappings differentiable at a point 10.3.2 The general rules for differentiation 10.3.3 Some examples 10.3.4 The partial deriatives of a mapping 10.3.5 Problems and exercises 10.4 The mean-value theorem and some examples of its use 10.4.1 The mean-value theorem 10.4.2 Some applications of the mean-value theorem 10.4.3 Problems and exercises 10.5 Higher-order derivatives 10.5.1 Definition of the nth differential 10.5.2 The derivative with respect to a vector and the computation of the values of the nth differential. 10.5.3 Symmetry of the higher-order differentials 10.5.4 Some remarks 10.5.5 Problems and exercises 10.6 Taylor's formula and methods of finding extrema 10.6.1 Taylor's formula for mappings 10.6.2 Methods of finding interior extrema 10.6.3 Some examples 10.6.4 Problems and e