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The second volume expounds classical analysis as it is today, as a part of unified mathematics, and its interactions with modern mathematical courses such as algebra, differential geometry, differential equations, complex and functional analysis. The book provides a firm foundation for advanced work in any of these directions.

Mathematical Analysis II. Vol.2

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