Intro
Preface
Contents
List of Symbols
1 Convergence of Sequences of Functions
1.1 Preliminaries and Notation
1.2 Pointwise and Uniform Convergence
1.3 Series of Functions
1.3.1 Power Series in the Complex Plane
1.3.2 Fourier Series
1.3.2.1 Dirichlet Kernel
1.3.2.2 Cesàro Means: Féjer Kernel
1.3.2.3 Poisson Kernel
1.3.3 Dirichlet Series
1.4 Exercises
References
2 Locally Convex Spaces
2.1 Topological Preliminaries
2.1.1 Basic Definitions
2.1.2 Metric and Normed Spaces
2.2 Seminorms
2.2.1 Locally Convex Topology
2.2.2 Continuity

2.2.3 Metrizable Locally Convex Spaces
2.3 The Dual of a Locally Convex Space
2.4 Examples of Spaces
2.4.1 Space of Continuous Functions
2.4.2 Köthe Echelon Spaces
2.5 Normable Spaces
2.6 Two Theorems on Spaces of Continuous Functions
2.6.1 Stone-Weierstraß Theorem
2.6.2 Ascoli Theorem
2.7 A Short Introduction to Hilbert Spaces
2.8 Exercises
References
3 Duality and Linear Operators
3.1 Hyperplanes
3.2 The Hahn-Banach Theorem
3.2.1 Analytic Version
3.2.2 Separation Theorems
3.2.3 Finite Dimensional Locally Convex Spaces
3.2.4 Banach Limits

3.3 Weak Topologies
3.4 The Bipolar Theorem
3.5 The Mackey-Arens Theorem
3.6 The Banach-Steinhaus Theorem
3.7 The Banach-Schauder Theorem
3.8 Topologies on the Space of Continuous Linear Mappings
3.9 Transpose of an Operator
3.10 Exercises
References
4 Spaces of Holomorphic and Differentiable Functions and Operators Between Them
4.1 Space of Holomorphic Functions
4.1.1 Locally Convex Structure
4.1.2 Representation as a Sequence Space
4.1.3 Montel Theorem
4.1.4 Dual of the Space of Entire Functions
4.2 Spaces of Differentiable Functions

4.3 Some Operators on Spaces of Functions
4.4 Exercises
References
5 Transitive and Mean Ergodic Operators
5.1 Transitive Operators
5.2 Mean Ergodic Operators
5.3 Examples
5.3.1 The Backward Shift
5.3.2 Composition Operators
5.3.3 Multiplication and Integration Operators
5.3.4 Differential Operators
5.4 Exercises
References
6 Schwartz Distributions and Linear Partial Differential Operators
6.1 Test Functions and Distributions
6.1.1 Definition and Examples
6.1.2 Differentiation of Distributions

6.1.3 Multiplication of a Distribution by a C∞-Function
6.1.4 Support of a Distribution and Distributions with Compact Support
6.2 The Space of Rapidly Decreasing Functions
6.3 Fourier Transform on S( RN)
6.4 Tempered Distributions and the Fourier Transform
6.5 Linear Partial Differential Operators
6.5.1 Fundamental Solutions. The Malgrange-Ehrenpreis Theorem
6.5.2 Solutions of Linear PDEs
6.6 Exercises
References
References
Index