Designed for students of probability and stochastic processes, as well as for students of functional analysis, specifically, this volume presents some chosen parts of functional analysis that can help clarify probability and stochastic processes. The subjects range from basic Hilbert and Banach spaces, through weak topologies and Banach algebras, to the theory of semigroups of bounded linear operators. Numerous standard and non-standard examples and exercises make the book suitable as a course textbook or for self-study.
Author(s): Adam Bobrowski
Publisher: Cambridge University Press
Year: 2005
Language: English
Pages: 407
Tags: Математика;Функциональный анализ;
Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 13
1.1 Elements of topology......Page 15
1.2 Measure theory......Page 17
1.3 Functions of bounded variation. Riemann–Stieltjes integral......Page 31
1.4 Sequences of independent random variables......Page 37
1.5 Convex functions. Hölder and Minkowski inequalities......Page 43
1.6 The Cauchy equation......Page 47
2.1 Linear spaces......Page 51
2.2 Banach spaces......Page 58
2.3 The space of bounded linear operators......Page 77
3.1 Projections in Hilbert spaces......Page 94
3.2 Definition and existence of conditional expectation......Page 101
3.3 Properties and examples......Page 105
3.4 The Radon–Nikodym Theorem......Page 115
3.5 Examples of discrete martingales......Page 117
3.6 Convergence of self-adjoint operators......Page 120
3.7 ... and of martingales......Page 126
4 Brownian motion and Hilbert spaces......Page 135
4.1 Gaussian families & the definition of Brownian motion......Page 137
4.2 Complete orthonormal sequences in a Hilbert space......Page 141
4.3 Construction and basic properties of Brownian motion......Page 147
4.4 Stochastic integrals......Page 153
5 Dual spaces and convergence of probability measures......Page 161
5.1 The Hahn–Banach Theorem......Page 162
5.2 Form of linear functionals in specific Banach spaces......Page 168
5.3 The dual of an operator......Page 176
5.4 Weak and weak∗ topologies......Page 180
5.5 The Central Limit Theorem......Page 189
5.6 Weak convergence in metric spaces......Page 192
5.7 Compactness everywhere......Page 198
5.8 Notes on other modes of convergence......Page 212
6.1 Banach algebras......Page 215
6.2 The Gelfand transform......Page 220
6.3 Examples of Gelfand transform......Page 222
6.4 Examples of explicit calculations of Gelfand transform......Page 231
6.5 Dense subalgebras of C(S)......Page 236
6.6 Inverting the abstract Fourier transform......Page 238
6.7 The Factorization Theorem......Page 245
7.1 The Banach–Steinhaus Theorem......Page 248
7.2 Calculus of Banach space valued functions......Page 252
7.3 Closed operators......Page 254
7.4 Semigroups of operators......Page 260
7.5 Brownian motion and Poisson process semigroups......Page 279
7.6 More convolution semigroups......Page 284
7.7 The telegraph process semigroup......Page 294
7.8 Convolution semigroups of measures on semigroups......Page 300
8.1 Semigroups of operators related to Markov processes......Page 308
8.2 The Hille–Yosida Theorem......Page 323
8.3 Generators of stochastic processes......Page 341
8.4 Approximation theorems......Page 354
9.1 Bibliographical notes......Page 377
9.2 Solutions and hints to exercises......Page 380
9.3 Some commonly used notations......Page 397
References......Page 399
Index......Page 404
