Function Spaces and Operators between them

The aim of this work is to present, in a unified and reasonably self-contained way, certain aspects of functional analysis which are needed to treat function spaces whose topology is not derived from a single norm, their topological duals and operators between those spaces. We treat spaces of continuous, analytic and smooth functions as well as sequence spaces. Operators of differentiation, integration, composition, multiplication and partial differential operators between those spaces are studied. A brief introduction to Laurent Schwartz’s theory of distributions and to Lars Hörmander’s approach to linear partial differential operators is presented. The novelty of our approach lies mainly on two facts. First of all, we show all these topics together in an accessible way, stressing the connection between them. Second, we keep it always at a level that is accessible to beginners and young researchers. Moreover, parts of the book might be of interest for researchers in functional analysis and operator theory. Our aim is not to build and describe a whole, complete theory, but to serve as an introduction to some aspects that we believe are interesting. We wish to guide any reader that wishes to enter in some of these topics in their first steps. Our hope is that they learn interesting aspects of functional analysis and become interested to broaden their knowledge about function and sequence spaces and operators between them. The text is addressed to students at a master level, or even undergraduate at the last semesters, since only knowledge on real and complex analysis is assumed. We have intended to be as self-contained as possible, and wherever an external citation is needed, we try to be as precise as we can. Our aim is to be an introduction to topics in, or connected with, different aspects of functional analysis. Many of them are in some sense classical, but we tried to show a unified direct approach; some others are new. This is why parts of these lectures might be of some interest even for researchers in related areas of functional analysis or operator theory. There is a full chapter about transitive and mean ergodic operators on locally convex spaces. This material is new in book form. It is a novel approach and can be of interest for researchers in the area. Preface 6 Contents 9 List of Symbols 12 1 Convergence of Sequences of Functions 15 1.1 Preliminaries and Notation 15 1.2 Pointwise and Uniform Convergence 16 1.3 Series of Functions 23 1.3.1 Power Series in the Complex Plane 24 1.3.2 Fourier Series 25 1.3.2.1 Dirichlet Kernel 30 1.3.2.2 Cesàro Means: Féjer Kernel 32 1.3.2.3 Poisson Kernel 35 1.3.3 Dirichlet Series 38 1.4 Exercises 48 References 51 2 Locally Convex Spaces 53 2.1 Topological Preliminaries 53 2.1.1 Basic Definitions 53 2.1.2 Metric and Normed Spaces 55 2.2 Seminorms 61 2.2.1 Locally Convex Topology 62 2.2.2 Continuity 64 2.2.3 Metrizable Locally Convex Spaces 69 2.3 The Dual of a Locally Convex Space 71 2.4 Examples of Spaces 74 2.4.1 Space of Continuous Functions 74 2.4.2 Köthe Echelon Spaces 76 2.5 Normable Spaces 81 2.6 Two Theorems on Spaces of Continuous Functions 83 2.6.1 Stone–Weierstraß Theorem 83 2.6.2 Ascoli Theorem 87 2.7 A Short Introduction to Hilbert Spaces 90 2.8 Exercises 100 References 102 3 Duality and Linear Operators 104 3.1 Hyperplanes 104 3.2 The Hahn–Banach Theorem 106 3.2.1 Analytic Version 106 3.2.2 Separation Theorems 108 3.2.3 Finite Dimensional Locally Convex Spaces 112 3.2.4 Banach Limits 114 3.3 Weak Topologies 117 3.4 The Bipolar Theorem 123 3.5 The Mackey–Arens Theorem 126 3.6 The Banach–Steinhaus Theorem 129 3.7 The Banach-Schauder Theorem 135 3.8 Topologies on the Space of Continuous Linear Mappings 138 3.9 Transpose of an Operator 144 3.10 Exercises 146 References 147 4 Spaces of Holomorphic and Differentiable Functions and Operators Between Them 150 4.1 Space of Holomorphic Functions 150 4.1.1 Locally Convex Structure 150 4.1.2 Representation as a Sequence Space 155 4.1.3 Montel Theorem 157 4.1.4 Dual of the Space of Entire Functions 158 4.2 Spaces of Differentiable Functions 163 4.3 Some Operators on Spaces of Functions 166 4.4 Exercises 170 References 171 5 Transitive and Mean Ergodic Operators 173 5.1 Transitive Operators 174 5.2 Mean Ergodic Operators 182 5.3 Examples 194 5.3.1 The Backward Shift 194 5.3.2 Composition Operators 198 5.3.3 Multiplication and Integration Operators 206 5.3.4 Differential Operators 208 5.4 Exercises 210 References 213 6 Schwartz Distributions and Linear Partial Differential Operators 214 6.1 Test Functions and Distributions 214 6.1.1 Definition and Examples 214 6.1.2 Differentiation of Distributions 223 6.1.3 Multiplication of a Distribution by a C∞-Function 225 6.1.4 Support of a Distribution and Distributions with Compact Support 228 6.2 The Space of Rapidly Decreasing Functions 230 6.3 Fourier Transform on S( RN) 233 6.4 Tempered Distributions and the Fourier Transform 241 6.5 Linear Partial Differential Operators 252 6.5.1 Fundamental Solutions. The Malgrange–Ehrenpreis Theorem 253 6.5.2 Solutions of Linear PDEs 261 6.6 Exercises 266 References 270 References 271 Index 275