Basic Analysis III: Mappings on Infinite Dimensional Spaces

✍ Scribed by James K. Peterson

• Publisher: Chapman and Hall/CRC
• Year: 2020
• Tongue: English
• Leaves: 469
• Edition: 1
• Category: Library

✦ Synopsis

Basic Analysis III: Mappings on Infinite Dimensional Spaces is intended as a first course in abstract linear analysis. This textbook cover metric spaces, normed linear spaces and inner product spaces, along with many other deeper abstract ideas such a completeness, operators and dual spaces. These topics act as an important tool in the development of a mathematically trained scientist.

Feature:

• Can be used as a traditional textbook as well as for self-study

• Suitable for undergraduates in mathematics and associated disciplines

• Emphasizes learning how to understand the consequences of assumptions using a variety of tools to provide the proofs of propositions

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Dedication Page
Table of Contents
I Introduction
1 Introduction
1.1 The Analysis Courses
1.1.1 Senior Level Analysis
1.1.2 The Graduate Analysis Courses
1.1.3 More Advanced Courses
1.2 Table of Contents
II Metric Spaces
2 Metric Spaces
2.1 The Construction of the Reals
2.1.1 Totally Ordering of the Rationals
2.1.2 The Construction of the Reals
2.1.3 The Equivalence Classes of Q are Totally Ordered
2.1.4 The Ordering and the Operations are Compatible
2.1.5 Cauchy Sequences of Equivalence Classes Converge
2.2 The Idea of a Metric
2.3 Examples
2.3.1 Homework
2.4 More on Symbol Sequences
2.4.1 Homework
2.5 Symbol Spaces over Two Symbols
2.5.1 Homework
2.6 Periodic Points of a Map
2.7 Completeness
2.7.1 Homework
2.8 Function Metric Spaces
2.9 Sequences and Series of Complex Numbers
2.10 Sequence Metric Spaces
2.10.1 Sequence Space Metrics
2.11 Hölder’s and Minkowski’s Inequality in Function Spaces
2.11.1 Function Space Metrics
2.12 More Completeness Results
2.13 More on Separability
3 Completing a Metric Space
3.1 The Completion of a Metric Space
3.2 Completing the Integrable Functions
III Normed Linear Spaces
4 Vector Spaces
4.1 Vector Spaces over a Field
4.1.1 The Span of a Set of Vectors
4.2 Every Vector Space Has a Basis
5 Normed Linear Spaces
5.1 Norms
5.1.1 Sequence Space Norms
5.1.2 Function Space Norms
5.2 The Schauder Basis
5.2.1 Schauder Basis Examples
5.3 The Linear Combination Theorem
5.4 Compactness
6 Linear Operators on Normed Spaces
6.1 Linear Transformations between Normed Linear Spaces
6.1.1 Basic Properties
6.2 Input - Output Ratios
6.3 Linear Operators between Normed Linear Spaces
6.4 Linear Operators on Rn to Rm
6.5 Eigenvalues and Eigenvectors for Operators
6.6 A Differential Operator Example
6.6.1 The Separation Constant is Positive
6.6.2 Case II: The Separation Constant is Zero
6.6.3 Case III: The Separation Constant is Negative
6.6.4 Homework
6.7 Spaces of Linear Operators
IV Inner Product Spaces
7 Inner Product Spaces
7.1 Inner Products
7.2 Hermitian Matrices
7.2.1 Constructing Eigenvalues
7.2.2 What Does This Mean?
7.3 Examples of Hilbert Spaces
7.4 Completing the Integrable Functions
7.5 Properties of Inner Product Spaces
8 Hilbert Spaces
8.1 Completeness and Projections
8.2 Projections and Consequences
8.3 Orthonormal Sequences
8.4 Fourier Coefficients and Completeness
9 Dual Spaces
9.1 Linear Functio
9.2 Weak Convergence
9.2.1 The Dual of l1 Sequences
9.2.2 The Dual of lp Sequences for Finite p
10 Hahn - Banach Results
10.1 Linear Extensions
10.2 The Hahn - Banach Theorem
10.3 Consequences
11 More About Dual Spaces
11.1 Reflexive Spaces
11.1.1 Homework
11.2 The Dual of the Continuous Functions
11.2.1 A Quick Look at Riemann - Stieljes Integration
11.2.2 Characterizing the Dual of the Set of Continuous Functions
11.2.3 A Norm Isometry for the Dual of the Continuous Functions
11.3 Riesz’s Characterization of the Hilbert Space Dual
11.4 Sesquilinear Forms
11.5 Adjoints on Normed Linear Spaces
12 Some Classical Results
12.1 First and Second Category Metric Spaces
12.1.1 The Uniform Boundedness Theorem
12.1.2 Some Fourier Series Do Not Converge Pointwise
12.2 The Open Mapping Theorem
12.2.1 Homework
12.3 The Closed Graph Theorem
12.3.1 Homework
V Operators
13 Stürm - Liouville Operators
13.1 ODE Background
13.2 The Stürm - Liouville Models
13.3 Properties
13.4 Linear Independence of the Solutions
13.5 Eigenvalue Behavior
13.6 The Inverse of the St¨urm - Liouville Differential Operator
13.6.1 The Actual Inversion
13.6.2 Verifying the Solution
13.6.3 More on the Eigenvalues
13.7 A Bessel’s Equation Example
13.7.1 The Bessel Function Code Implementation
13.7.2 Approximation with the Bessel Functions
14 Self-Adjoint Operators
14.1 Integral Operators
14.1.1 Integral Operator with Symmetric and Hermitian Kernels
14.1.2 Properties of the Integral Operator
14.1.3 L is Well-Defined
14.1.4 Equicontinuous Families Determined by the Kernel
14.1.5 Characterization of the Self-Adjoint Operator Norm
14.2 Eigenvalues of Self-Adjoint Operators
14.3 Back to the Stürm - Liouville Operator
14.3.1 Derivative Boundary Conditions
14.3.2 State Boundary Conditions
14.3.3 Completeness for the Stürm - Liouville Eigenfunction Sequences
14.4 The Ball and Stick Model
14.4.1 Completeness
VI Topics in Applied Modeling
15 Fields and Charges on a Set
15.1 Rings and Fields of Subsets
15.2 Charges
15.3 Ordered Vector Spaces and Lattices
15.4 The Structure of Bounded Charges
15.5 Measures
16 Games
16.1 Finite Numbers of Players
16.2 Games of Transferable Utility
16.3 Payoff Vectors as Charges
16.3.1 Countably Infinite Numbers of Players
16.4 Some Additional Dual Spaces
16.4.1 The Dual of Sequences That Converge
16.4.2 The Dual of Sequences That Converge to 0
16.5 A Digression to Integration Theory
16.5.1 A Riemann Integral Extension
16.6 The Dual of Bounded Measurable Functions
16.7 Connections to Game Theory
VII Summing It All Up
17 Summing It All Up
VIII References
IX Detailed Index