Elementary Functional Analysis

by Marat V. Markin

Overview

While there is a plethora of excellent, but mostly "tell-it-all'' books on the subject, this one is intended to take a unique place in what today seems to be a still wide open niche for an introductory text on the basics of functional analysis to be taught within the existing constraints of the standard, for the United States, one-semester graduate curriculum (fifteen weeks with two seventy-five-minute lectures per week).　

The book consists of seven chapters and an appendix taking the reader from the fundamentals of abstract spaces (metric, vector, normed vector, and inner product), through the basics of linear operators and functionals, the three fundamental principles (the Hahn-Banach Theorem, the Uniform Boundedness Principle, the Open Mapping Theorem and its equivalents: the Inverse Mapping and Closed Graph Theorems) with their numerous profound implications and certain interesting applications, to the elements of the duality and reflexivity theory. Chapter 1 outlines some necessary preliminaries, while the Appendix gives a concise discourse on the celebrated Axiom of Choice, its equivalents (the Hausdorff Maximal Principle, Zorn's Lemma, and Zermello's Well-Ordering Principle), and ordered sets.　

Being designed as a text to be used in a classroom, the book constantly calls for the student's actively mastering the knowledge of the subject matter. It contains 112 Problems, which are indispensable for understanding and moving forward. Many important statements are given as problems, a lot of these are frequently referred to and used in the main body. There are also 376 Exercises throughout the text, including Chapter 1 and the Appendix, which require of the student to prove or verify a statement or an example, fill in necessary details in a proof, or provide an intermediate step or a counterexample. They are also an inherent part of the material. More difficult problems are marked with an asterisk, many problem and exercises being supplied with "existential'' hints.　

The book is generous on Examples and contains numerous Remarks accompanying every definition and virtually each statement to discuss certain subtleties, raise questions on whether the converse assertions are true, whenever appropriate, or whether the conditions are essential.　　

The prerequisites are set intentionally quite low, the students not being assumed to have taken graduate courses in real or complex analysis and general topology, to make the course accessible and attractive to a wider audience of STEM (science, technology, engineering, and mathematics) graduate students or advanced undergraduates with a solid background in calculus and linear algebra.

With proper attention given to applications, plenty of examples, problems, and exercises, this well-designed text is ideal for a one-semester graduate course on the fundamentals of functional analysis for students in mathematics, physics, computer science, and engineering.

Contents
Preliminaries
Metric Spaces
Normed Vector and Banach Spaces
Inner Product and Hilbert Spaces
Linear Operators and Functionals
Three Fundamental Principles of Linear Functional Analysis
Duality and Reflexivity
The Axiom of Choice and Equivalents

Product Details

ISBN-13:	9783110614091
Publisher:	De Gruyter
Publication date:	10/08/2018
Series:	De Gruyter Textbook
Sold by:	Barnes & Noble
Format:	eBook
Pages:	330
File size:	44 MB
Note:	This product may take a few minutes to download.
Age Range:	18 Years

About the Author

Marat V. Markin, California State University, Fresno, USA.

Table of Contents

Preface vii

1 Preliminaries 1

1.1 Set Theoretic Basics 1

1.1.1 Some Terminology and Notations 1

1.1.2 Cardinality and Countability 2

1.2 Terminology Related to Functions 4

1.3 Upper and Lower Limits 6

2 Metric Spaces 7

2.1 Definition and Examples 7

2.2 Hölder's and Minkowski's Inequalities 9

2.2.1 Conjugate Indices 9

2.2.2 Young's inequality 9

2.2.3 The Case of n-Tuples 10

2.2.4 Sequential Case 12

2.3 Subspaces of a Metric Space 14

2.4 Function Spaces 14

2.5 Further Properties of Metric 16

2.6 Convergence and Continuity 17

2.6.1 Convergence of a Sequence 17

2.6.2 Continuity, Uniform Continuity, and Lipschitz Continuity 18

2.7 Balls, Separation, and Boundedness 20

2.8 Interior Points, Open Sets 23

2.9 Limit Points, Closed Sets 24

2.10 Dense Sets and Separable Spaces 27

2.11 Exterior and Boundary 28

2.12 Equivalent Metrics, Homeomorphisms and Isometries 29

2.12.1 Equivalent Metrics 29

2.12.2 Homeomorphisms and Isometries 30

2.13 Completeness and Completion 31

2.13.1 Cauchy/Fundamental Sequences 31

2.13.2 Complete Metric Spaces 33

2.13.3 Subspaces of Complete Metric Spaces 37

2.13.4 Nested Balls Theorem 37

2.13.5 Completion 40

2.14 Category and Baire Category Theorem 43

2.14.1 Nowhere Denseness 43

2.14.2 Category 46

2.14.3 Baire Category Theorem 47

2.15 Compactness 49

2.15.1 Total Boundedness 50

2.15.2 Compactness, Precompactness 54

2.15.3 Hausdorff Criterion 60

2.15.4 Compactness in Certain Complete Metric Spaces 62

2.15.5 Other Forms of Compactness 64

2.15.6 Equivalence of Different Forms of Compactness 65

2.15.7 Compactness and Continuity 67

2.16 Space (C(X, Y), p∞) 70

2.17 Arzelà-Ascoll Theorem 72

2.17.1 Uniform Boundedness and Equicontinuity 72

2.17.2 Arzelà-Ascoli Theorem 73

2.17.3 Application: Peano's Existence Theorem 76

2.18 Stone-Weierstrass Theorem 79

2.18.1 Weierstrass Approximation Theorem 79

2.18.2 Algebras 79

2.18.3 Stone-Weierstrass Theorem 82

2.18.4 Applications 87

2.19 Problems 87

3 Normed Vector and Banach Spaces 97

3.1 Vector Spaces 97

3.1.1 Definition, Examples, Properties 97

3.1.2 Homomorphismsand Isomorphisms 100

3.1.3 Subspaces 101

3.1.4 Spans and Linear Combinations 103

3.1.5 Linear Independence, Hamel Bases, Dimension 103

3.1.6 New Spaces from Old 108

3.1.7 Disjoint and Complementary Subspaces, Direct Sum Decompositions, Deficiency and Codimension 111

3.2 Normed Vector and Banach Spaces 114

3.2.1 Definitions and Examples 114

3.2.2 Series and Completeness Characterization 118

3.2.3 Comparing Norms, Equivalent Norms 119

3.2.4 Isometric Isomorphisms 120

3.2.5 Completion 121

3.2.6 Topological and Schauder Bases 122

3.3 Finite-Dimensional Spaces and Related Topics 125

3.3.1 Norm Equivalence and Completeness 125

3.3.2 Finite-Dimensional Subspaces and Bases of Banach Spaces 127

3.4 Riesz's Lemma and Implications 130

3.5 Convexity, Strictly Convex Normed Vector Spaces 132

3.5.1 Convexity 132

3.5.2 Strictly Convex Normed Vector Spaces 133

3.6 Problems 137

4 Inner Product and Hilbert Spaces 141

4.1 Definitions and Examples 141

4.2 Inner Product Norm, Cauchy-Schwarz Inequality 143

4.3 Hilbert Spaces 145

4.4 Certain Geometric Properties 147

4.4.1 Polarization Identities 147

4.4.2 Parallelogram Law 147

4.4.3 Orthogonality 151

4.5 Nearest Point Property 152

4.6 Projection Theorem 155

4.6.1 Orthogonal Complements 155

4.6.2 Projection Theorem 157

4.7 Completion 159

4.8 Gram Determinant 160

4.9 Orthogonal and Orthonormal Sets 165

4.10 Gram-Schmidt Process 170

4.11 Generalized Fourier Series 171

4.11.1 Finite Orthonormal Set 172

4.11.2 Arbitrary Orthonormal Set 174

4.11.3 Orthonormal Sequence 178

4.12 Orthonormal Bases and Orthogonal Dimension 179

4.13 Problems 184

5 Linear Operators and Functionals 187

5.1 Linear Operators and Functionals 187

5.1.1 Definitions and Examples 187

5.1.2 Kernel, Range, and Graph 189

5.1.3 Rank-Nullity and Extension Theorems 189

5.2 Bounded Linear Operators and Functionals 192

5.2.1 Definitions, Properties, and Examples 192

5.2.2 Space of Bounded Linear Operators, Dual Space 197

5.3 Closed Linear Operators 201

5.4 Problems 204

6 Three Fundamental Principles of Linear Functional Analysis 207

6.1 Hahn-Banach Theorem 207

6.1.1 Hahn-Banach Theorem for Real Vector Spaces 207

6.1.2 Hahn-Banach Theorem for Normed Vector Spaces 210

6.2 Implications of the Hahn-Banach Theorem 212

6.2.1 Separation and Norm Realization 212

6.2.2 Characterization of Fundamentally 215

6.2.3 Sufficiency for Separability 216

6.2.4 Isometric Embedding Theorems 217

6.2.5 Second Dual Space and Canonical Isomorphism 218

6.2.6 Closed Complemented Subspaces 219

6.3 Weak and Weak^* Convergence 221

6.4 Uniform Boundedness Principle, the Banach-Steinhaus Theorem 225

6.4.1 Uniform Boundedness Principle 225

6.4.2 Banach-Steinhaus Theorem 228

6.5 Applications of the Uniform Boundedness Principle 231

6.5.1 Weak Boundedness 232

6.5.2 Matrix Methods of Convergence and Summability 233

6.6 Open Mapping, inverse Mapping, and Closed Graph Theorems 240

6.6.1 Open Mapping Theorem 240

6.6.2 Inverse Mapping Theorem and Applications 245

6.6.3 Closed Graph Theorem and Application 249

6.6.4 Equivalence of OMT, IMT, and CGT 255

6.7 Problems 256

7 Duality and Reflexivity 261

7.1 Self-Duality of Hilbert Spaces 261

7.1.1 Riesz Representation Theorem 261

7.1.2 Linear Bounded Functionals on Certain Hilbert Spaces 264

7.1.3 Weak Convergence in Hilbert Spaces 264

7.2 Duality of Finite-Dimensional Spaces 265

7.2.1 Representation Theorem 265

7.2.2 Weak Convergence in Finite-Dimensional Spaces 268

7.3 Duality of Sequence Spaces 268

7.3.1 Representation Theorem for I^*_ρ 268

7.3.2 Weak Convergence in I_ρ (1 ≤ p < ∞) 272

7.3.3 Duality and Weak Convergence for (c₀, || ||_∞) 276

7.4 Duality and Weak Convergence for (C[a, b], || ||_∞) 277

7.4.1 Riesz Representation Theorem for C^* [a, b] 277

7.4.2 Weak Convergence in (C[a, b], || _∞) 278

7.5 Reflexivity 278

7.5.1 Definition and Examples 278

7.5.2 Completeness of a Reflexive Space 283

7.5.3 Reflexivity of a Closed Subspace 283

7.5.4 Isometric Isomorphism and Reflexivity 285

7.5.5 Characterization of Reflexivity 286

7.5.6 Weak Convergence and Weak Completeness 287

7.5.7 Bounded Sequence Property 288

7.6 Problems 290

A The Axiom of Choice and Equivalents 293

A.1 The Axiom of Choice 293

A.1.1 The Axiom of Choice 293

A.1.2 Controversy 293

A.1.3 Timeline 294

A.2 Ordered Sets 294

A.3 Equivalents 298

Bibliography 303

Index 305

Editorial Reviews

"While there is a plethora of excellent, but mostly "tell-it-all'' books on the subject, this one is intended to take a unique place in what today seems to be a still wide open niche for an introductory text on the basics of functional analysis to be taught within the existing constraints of the standard, for the United States, one-semester graduate curriculum (fifteen weeks with two seventy-five-minute lectures per week). [...] The book is generous on Examples and contains numerous Remarks accompanying every definition and virtually each statement to discuss certain subtleties, raise questions on whether the converse assertions are true, whenever appropriate, or whether the conditions are essential. The prerequisites are set intentionally quite low, the students not being assumed to have taken graduate courses in real or complex analysis and general topology, to make the course accessible and attractive to a wider audience of STEM (science, technology, engineering, and mathematics) graduate students or advanced undergraduates with a solid background in calculus and linear algebra. With proper attention given to applications, plenty of examples, problems, and exercises, this well-designed text is ideal for a one-semester graduate course on the fundamentals of functional analysis for students in mathematics, physics, computer science, and engineering. [...]"
In: Bookauthority, https://bookauthority.org/books/beginner-functional-analysis-books, Letzter Aufruf am 22.10.2019

„Der Autor stellt übersichtlich die wesentlichen Themen der klassischen Funktionalanalysis in einem angemessenen Umfang […] dar. […] Das Buch ist als eine geeignete Ergänzung zu einem entsprechenden Grundkurs Analysis […] zu verstehen."
Dr. Torsten Mergen, Universityät des Saarlandes

From the Publisher