Functional Analysis

Gerardo Chacón, Humberto Rafeiro, Juan Camilo Vallejo

• Publisher: De Gruyter
• Year: 2016
• eBook ISBN: 9783110433647
• Edition: 1
• Topic: Mathematics
• Subtopic: Functional Analysis
• Index: Mathematics

This textbook on functional analysis offers a short and concise introduction to the subject. The book is designed in such a way as to provide a smooth transition between elementary and advanced topics and its modular structure allows for an easy assimilation of the content. Starting from a dedicated chapter on the axiom of choice, subsequent chapters cover Hilbert spaces, linear operators, functionals and duality, Fourier series, Fourier transform, the fixed point theorem, Baire categories, the uniform bounded principle, the open mapping theorem, the closed graph theorem, the Hahn–Banach theorem, adjoint operators, weak topologies and reflexivity, operators in Hilbert spaces, spectral theory of operators in Hilbert spaces, and compactness. Each chapter ends with workable problems.
The book is suitable for graduate students, but also for advanced undergraduates, in mathematics and physics.

Contents:
List of Figures
Basic Notation
Choice Principles
Hilbert Spaces
Completeness, Completion and Dimension
Linear Operators
Functionals and Dual Spaces
Fourier Series
Fourier Transform
Fixed Point Theorem
Baire Category Theorem
Uniform Boundedness Principle
Open Mapping Theorem
Closed Graph Theorem
Hahn–Banach Theorem
The Adjoint Operator
Weak Topologies and Reflexivity
Operators in Hilbert Spaces
Spectral Theory of Operators on Hilbert Spaces
Compactness
Bibliography
Index

1Choice Principles

Learning Targets

✓ Introduction to the axiom of choice.

✓ Learn some choice principles which are equivalent to the axiom of choice.

✓ Get acquainted with and understand how to use some type of choice principles in a real-world situation.

1.1Axiom of Choice

The axiom of choice is a devise used when we need to iterate some process infinitely, e.g., when we need to choose some infinite elements from a set. One of the formulations of the axiom uses the notion of choice function and the others rely on the Cartesian product.

Definition 1.1 (Choice Function). Let X be a nonempty set. A function f : 2^X → X is said to be a choice function on the set X if f(A) ∈ A when ∅ ≠ A ⊆ X.

With the notion of choice function we can state the axiom of choice.

Axiom 1.2 (Axiom of Choice). For every nonempty set there exists a choice function.

The axiom of choice can be phrased as

For every family A of disjoint nonempty sets there exists a set B which has exactly one element in common with each set belonging to A.

Although this axiom seems to be true, at least for the case of finite choices or numerable choices, in the case of nonnumerable infinite choices, the situation is quite different. Even for the numerable choices it is somewhat an illusion, since we do not have a way to guarantee that we can choose all the numerable elements without resorting to some type of axiom of choice. For example, to guarantee the existence of the natural numbers ℕ it is necessary to assume some type of axiom of infinity, as is done in the Zermelo–Fraenkel set theory. The power of the axiom of choice lies in the fact that it permits to choose an infinite number of things instantaneously. In Ref. [4] we have the following remark:

Several mathematicians claimed that proofs involving the axiom of choice have a different nature from proofs not involving it, because the axiom of choice is a unique set theoretical principle which states the existence of a set without giving a method of defining (“constructing”) it, i.e. is not effective.

In the beginning of the twentieth century there were disputes among several renowned mathematicians regarding the acceptance of this axiom. One of the serious obstacles is the so-called Banach–Tarski paradox.

Banach–Tarski Paradox: The unit ball B := {(x,y ,z ) ∈ ℝ³ : x² + y² + z² ≤ 1} in three dimensions can be disassembl...

Table of contents

• Cover
• Title Page
• Copyright
• Preface
• Contents
• List of Figures
• Basic Notation
• 1 Choice Principles
• 2 Hilbert Spaces
• 3 Completeness, Completion and Dimension
• 4 Linear Operators
• 5 Functionals and Dual Spaces
• 6 Fourier Series
• 7 Fourier Transform
• 8 Fixed Point Theorem
• 9 Baire Category Theorem
• 10 Uniform Boundedness Principle
• 11 Open Mapping Theorem
• 12 Closed Graph Theorem
• 13 Hahn–Banach Theorem
• 14 The Adjoint Operator
• 15 Weak Topologies and Reflexivity
• 16 Operators in Hilbert Spaces
• 17 Spectral Theory of Operators on Hilbert Spaces
• 18 Compactness
• Bibliography
• Index