Table of Contents
Preface v
1 Vector, metric, normed and Banach spaces 1
1.1 Vector spaces 1
1.2 Metric spaces 24
1.3 Useful inequalities 29
1.4 Complete spaces 32
1.5 Normed spaces 39
1.6 Banach spaces 53
1.7 Inner product spaces 54
1.8 Hilbert spaces 62
1.9 Separable spaces 69
1.10 Advanced practical problems 70
2 Lebesgue integration 73
2.1 Lebesgue outer measure. Measurable sets 73
2.2 The Lebesgue measure. The Borel-Cantelli lemma 95
2.3 Nonmeasurable sets 99
2.4 The Cantor set. The Cantor-Lebesgue function 102
2.5 Lebesgue measurable functions 107
2.6 The Riemann integral 124
2,7 Lebesgue integration 125
2.7.1 The Lebesgue integral of a bounded measurable function over a set of finite measure 125
2.7.2 The Lebesgue integral of a measurable nonnegative function 135
2.7.3 The general Lebesgue integral 143
2.8 Continuity and differentiability of monotone functions. Lebesgue's theorem 158
2.9 General measure spaces 168
2.10 General measurable functions 169
2.11 Integration over general measure spaces 171
2.12 Advanced practical problems 176
3 The Lρ spaces 179
3.1 Definition 179
3.2 The inequalities of Holder and Minkowski 180
3.3 Some properties 182
3.4 The Riesz-Fischer theorem 183
3.5 Separability 189
3.6 Duality 190
3.7 General Lρ spaces 204
3.8 Advanced practical problems 207
4 Linear operators 209
4.1 Definition 209
4.2 Linear operators in normed vector spaces 211
4.3 Inverse operators 229
4.4 Advanced practical problems 233
5 Linear functionals 235
5.1 The Hahn-Banach extension theorem 235
5.2 The general form of the linear functionals En in the case F = R 241
5.3 The general form of the linear functionals on Hilbert spaces 242
5.4 Weak convergence of sequences of functionals 244
5.5 Advanced practical problems 244
6 Relatively compact sets in metric and normed spaces. Compact operators 247
6.1 Definitions. General theorems 247
6.2 Criteria for compactness of sets in metric spaces 250
6.3 A Criteria for relative compactness in the space C([a, b]) 254
6.4 A Criteria for compactness in the space Lρ ([a, b]), p > 1 257
6.5 Compact operators 260
6.6 Advanced practical problems 263
7 Self-adjoint operators in Hilbert spaces 265
7.1 Adjoint operators. Self-adjoint operators 265
7.2 Unitary operators 266
7.3 Projection operators 267
8 The method of the small parameter 273
8.1 Abstract functions of a real variable 273
8.2 Power series 280
8.3 Analytic abstract functions and Taylor's series 282
8.4 The method of the smaller parameter 286
8.5 An application to integral equations 289
9 The parameter continuation method 299
9.1 Statement of the basic result 299
9.2 An application to a boundary value problem for a class of second order ordinary differential equations 300
10 Fixed-point theorems and applications 309
10.1 The Banach fixed-point theorem 309
10.2 The Brinciari fixed-point theorem 311
10.3 The Brouwer fixed-point theorem 318
10.4 The Schauder fixed-point theorem 320
10.5 Non-compact Type Krasnoset'skii fixed-point theorems 322
10.6 Fixed-point results for the sum T + S 325
10.7 Fixed-point results to one parameter operator equations and eigenvalues problems 338
10.8 Application to perturbed Volterra integral equation 339
10.9 Application to transport equations 344
10.10 Application to a class of difference equations 348
10.11 Application to a Darboux problem 360
A Sets and mappings 375
A.1 Union and intersection of sets 375
A.2 Mappings between sets 376
A.3 Countable and uncountable sets 379
A.4 Continuous real-valued functions on a real variable 383
B Functions of bounded variation 385
Bibliography 389
Index 391
