Introduction to Modern Analysis

Description

This textbook provides an introduction to modern analysis aimed at advanced undergraduate and graduate-level students of mathematics. Professional academics will also find this to be a useful reference work. It covers measure theory, basic functional analysis, single operator theory, spectral theory of bounded and unbounded operators, semigroups of operators, and Banach algebras. Further, this new edition of the textbook also delves deeper into C*-algebras and their standard constructions, von Neumann algebras, probability and mathematical statistics, and partial differential equations. Most chapters contain relatively advanced topics alongside simpler ones, starting from the very basics of modern analysis and slowly advancing to more involved topics. The text is supplemented by many exercises, to allow readers to test their understanding and practical analysis skills. Cover Titlepage Copyright Dedication Contents Preface to the First Edition Preface to the Second Edition 1 Measures 1.1 Measurable sets and functions 1.2 Positive measures 1.3 Integration of non-negative measurable functions 1.4 Integrable functions 1.5 Lp-spaces 1.6 Inner product 1.7 Hilbert space: a first look 1.8 The Lebesgue–Radon–Nikodym theorem 1.9 Complex measures 1.10 Convergence 1.11 Convergence on finite measure space 1.12 Distribution function 1.13 Truncation Exercises 2 Construction of measures 2.1 Semi-algebras 2.2 Outer measures 2.3 Extension of measures on algebras 2.4 Structure of measurable sets 2.5 Construction of Lebesgue–Stieltjes measures 2.6 Riemann vs. Lebesgue 2.7 Product measure Exercises 3 Measure and topology 3.1 Partition of unity 3.2 Positive linear functionals 3.3 The Riesz–Markov representation theorem 3.4 Lusin's theorem 3.5 The support of a measure 3.6 Measures on Rk; differentiability Exercises 4 Continuous linear functionals 4.1 Linear maps 4.2 The conjugates of Lebesgue spaces 4.3 The conjugate of Cc(X) 4.4 The Riesz representation theorem 4.5 Haar measure Exercises 5 Duality 5.1 The Hahn–Banach theorem 5.2 Reflexivity 5.3 Separation 5.4 Topological vector spaces 5.5 Weak topologies 5.6 Extremal points 5.7 The Stone–Weierstrass theorem 5.8 Operators between Lebesgue spaces: Marcinkiewicz's interpolation theorem 5.9 Fixed points 5.10 The bounded weak*-topology Exercises 6 Bounded operators 6.1 Category 6.2 The uniform boundedness theorem 6.3 The open mapping theorem 6.4 Graphs 6.5 Quotient space 6.6 Operator topologies Exercises 7 Banach algebras 7.1 Basics 7.2 Commutative Banach algebras 7.3 Involutions and C*-algebras 7.4 Normal elements 7.5 The Arens products Exercises 8 Hilbert spaces 8.1 Orthonormal sets 8.2 Projections 8.3 Orthonormal bases 8.4 Hilbert dimension 8.5 Isomorphism of Hilbert spaces 8.6 Direct sums 8.7 Canonical model 8.8 Tensor products 8.8.1 An interlude: tensor products of vector spaces 8.8.2 Tensor products of Hilbert spaces Exercises 9 Integral representation 9.1 Spectral measure on a Banach subspace 9.2 Integration 9.3 Case Z=X 9.4 The spectral theorem for normal operators 9.5 Parts of the spectrum 9.6 Spectral representation 9.7 Renorming method 9.8 Semi-simplicity space 9.9 Resolution of the identity on Z 9.10 Analytic operational calculus 9.11 Isolated points of the spectrum 9.12 Compact operators Exercises 10 Unbounded operators 10.1 Basics 10.2 The Hilbert adjoint 10.3 The spectral theorem for unbounded selfadjoint operators 10.4 The operational calculus for unbounded selfadjoint operators 10.5 The semi-simplicity space for unbounded operators in Banach space 10.6 Symmetric operators in Hilbert space 10.7 Quadratic forms Exercises 11 C*-algebras 11.1 Notation and examples 11.2 The continuous operational calculus continued 11.3 Positive elements 11.4 Approximate identities 11.5 Ideals 11.6 Positive linear functionals 11.7 Representations and the Gelfand–Naimark–Segal construction 11.7.1 Irreducible representations 11.8 Positive linear functionals and convexity 11.8.1 Pure states 11.8.2 Decompositions of functionals Exercises 12 Von Neumann algebras 12.1 Preliminaries 12.2 Commutants 12.3 Density 12.4 The polar decomposition 12.5 W*-algebras 12.6 Hilbert–Schmidt and trace-class operators 12.7 Commutative von Neumann algebras 12.8 The enveloping von Neumann algebra of a C*-algebra Exercises 13 Constructions of C*-algebras 13.1 Tensor products of C*-algebras 13.1.1 Tensor products of algebras 13.1.2 Tensor products of C*-algebras throughrepresentations 13.1.3 The maximal tensor product 13.1.4 Tensor products of bounded linear functionals 13.1.5 The minimal tensor product 13.1.6 Tensor products by commutative C*-algebras 13.2 Group C*-algebras 13.2.1 Unitary representations 13.2.2 The definition and representations of the group C*-algebra 13.2.3 Properties of the group C*-algebra Exercises Application I Probability I.1 Heuristics I.2 Probability space I.2.1 L2-random variables I.3 Probability distributions I.4 Characteristic functions I.5 Vector-valued random variables I.6 Estimation and decision I.6.1 Confidence intervals I.6.2 Testing of hypothesis and decision I.6.3 Tests based on a statistic I.7 Conditional probability I.7.1 Heuristics I.7.2 Conditioning by an r.v. I.8 Series of L2 random variables I.9 Infinite divisibility I.10 More on sequences of random variables Application II Distributions II.1 Preliminaries II.2 Distributions II.3 Temperate distributions II.3.1 The spaces Wp,k II.4 Fundamental solutions II.5 Solution in E II.6 Regularity of solutions II.7 Variable coefficients II.8 Convolution operators II.9 Some holomorphic semigroups Bibliography Index