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醉染图书现代分析及其应用教程9787560391021
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Preface
1 Prelude to Modern Analysis
1.1 Introduction
1.2 Sets and numbers
1.3 Functions or mappings
1.4 Countability
1.5 Point sets
1.6 Open and closed sets
1.7 Sequences
1.8 Series
1.9 Functions of a real variable
1.10 Uniform convergence
1.11 Some linear algebra
1.12 Setting off
2 Metric Spaces
2.1 Definition of a metric space
2.2 Examples of metric spaces
. Solved problems
2.4 Exercises
2.5 Convergence in a metric space
2.6 Examples on comleess
2.7 Subspace of a metric space
2.8 Solved problems
2.9 Exercises
3 The Fixed Point Theorem and its Applications
3.1 Mappings between metric spaces
3.2 The fixed point theorem
3.3 Applications
3.4 Perturbation mappings
3.5 Exercises
4 Compactness
4.1 Compact sets
4.2 Ascolis theorem
4.3 Application to approximation theory
4.4 Solved problems
4.5 Exercises
5 Topological Spaces
5.1 Definitions and examples
5.2 Closed sets
5.3 Compact sets
5.4 Continuity in topological spaces
5.5 Homeomorphisms; connectedness
5.6 Solved problems
5.7 Exercises
6 Normed Vector Spaces
6.1 Definition of a normed vector space; examples
6.2 Convergence in normed spaces
6.3 Solved problems
6.4 Exercises
6.5 Finite-dimensional normed vector spaces
6.6 Some approximation theory
6.7 Chebyshev theory
6.8 The Weierstrass approximation theorem
6.9 Solved problems
6.10 Exercises
7 Mappings on Normed Spaces
7.1 Bounded linear mappings
7.2 Norm of an operator
7.3 Functionals
7.4 Solved problems
7.5 Exercises
7.6 Inverse mappings
7.7 Application to integral equations
7.8 Application to numerical analysis
7.9 Exercises
7.10 Unbounded mappings
8 Inner Product Spaces
8.1 Definitions; simple consequences
8.2 Orthonormal vectors
8.3 Least squares approximation
8.4 The Riesz representation theorem
8.5 Solved problems
8.6 Exercises
9 Hilbert Space
9.1 Definition of Hilbert space
9.2 The adjoint operator
9.3 Separability
9.4 Solved problems
9.5 Exercises
9.6 Complete orthonormal sets; generalised Fourier series
9.7 Hilberspceismorphism
9.8 Exercises
Bibliography
Selected Solutions
Index
编辑手记
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