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[资源] Cambridge2003年A Course in Modern Analysis and Its Applications

Preface page ix
1 Prelude to Modern Analysis 1
1.1 Introduction 1
1.2 Sets and numbers 3
1.3 Functions or mappings 10
1.4 Countability 14
1.5 Point sets 20
1.6 Open and closed sets 28
1.7 Sequences 32
1.8 Series 44
1.9 Functions of a real variable 52
1.10 Uniform convergence 59
1.11 Some linear algebra 69
1.12 Setting off 83
2 Metric Spaces 84
2.1 Definition of a metric space 84
2.2 Examples of metric spaces 85
2.3 Solved problems 94
2.4 Exercises 96
2.5 Convergence in a metric space 98
2.6 Examples on completeness 102
2.7 Subspace of a metric space 107
2.8 Solved problems 109
2.9 Exercises 111
3 The Fixed Point Theorem and its Applications 113
3.1 Mappings between metric spaces 113
3.2 The fixed point theorem 115
v
vi Contents
3.3 Applications 118
3.4 Perturbation mappings 130
3.5 Exercises 135
4 Compactness 140
4.1 Compact sets 140
4.2 Ascoli’s theorem 145
4.3 Application to approximation theory 149
4.4 Solved problems 152
4.5 Exercises 153
5 Topological Spaces 155
5.1 Definitions and examples 155
5.2 Closed sets 158
5.3 Compact sets 160
5.4 Continuity in topological spaces 164
5.5 Homeomorphisms; connectedness 167
5.6 Solved problems 170
5.7 Exercises 171
6 Normed Vector Spaces 174
6.1 Definition of a normed vector space; examples 174
6.2 Convergence in normed spaces 178
6.3 Solved problems 181
6.4 Exercises 185
6.5 Finite-dimensional normed vector spaces 187
6.6 Some approximation theory 192
6.7 Chebyshev theory 195
6.8 The Weierstrass approximation theorem 199
6.9 Solved problems 205
6.10 Exercises 208
7 Mappings on Normed Spaces 210
7.1 Bounded linear mappings 210
7.2 Norm of an operator 214
7.3 Functionals 218
7.4 Solved problems 221
7.5 Exercises 223
7.6 Inverse mappings 225
7.7 Application to integral equations 229
7.8 Application to numerical analysis 235
7.9 Exercises 243
7.10 Unbounded mappings 246
Contents vii
8 Inner Product Spaces 251
8.1 Definitions; simple consequences 251
8.2 Orthonormal vectors 259
8.3 Least squares approximation 267
8.4 The Riesz representation theorem 273
8.5 Solved problems 275
8.6 Exercises 278
9 Hilbert Space 281
9.1 Definition of Hilbert space 281
9.2 The adjoint operator 282
9.3 Separability 288
9.4 Solved problems 291
9.5 Exercises 292
9.6 Complete orthonormal sets; generalised Fourier series 294
9.7 Hilbert space isomorphism 303
9.8 Exercises 307
Bibliography 309
Selected Solutions 312
Index 330

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附件 1 : A_Course_in_Modern_Analysis_and_Its_Applications__Australian_MATH_Society_Lecture_Series)_-__Graeme_L._Cohen_-__(_Cambridge_University_Press_-__2003_-_pp.349_).pdf

2015-03-01 15:06:32, 1.72 M