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Basic Analysis V: Functional Analysis and Topology

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Basic Analysis V: Functional Analysis and Topology introduces graduate students in science to concepts from topology and functional analysis, both linear and nonlinear. It is the fifth book in a series designed to train interested readers how to think properly using mathematical abstractions, and how to use the tools of mathematical analysis in applications.

It is important to realize that the most difficult part of applying mathematical reasoning to a new problem domain is choosing the underlying mathematical framework to use on the problem. Once that choice is made, we have many tools we can use to solve the problem. However, a different choice would open up avenues of analysis from a different, perhaps more productive, perspective.

In this volume, the nature of these critical choices is discussed using applications involving the immune system and cognition.

Features

  • Develops a proof of the Jordan Canonical form to show some basic ideas in algebraic topology

  • Provides a thorough treatment of topological spaces, finishing with the Krein–Milman theorem

  • Discusses topological degree theory (Brouwer, Leray–Schauder, and Coincidence)

  • Carefully develops manifolds and functions on manifolds ending with Riemannian metrics

  • Suitable for advanced students in mathematics and associated disciplines

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

Author

James K. Peterson is an Emeritus Professor at the School of Mathematical and Statistical Sciences, Clemson University.

He tries hard to build interesting models of complex phenomena using a blend of mathematics, computation, and science. To this end, he has written four books on how to teach such things to biologists and cognitive scientists. These books grew out of his Calculus for Biologists courses offered to the biology majors from 2007 to 2015.

He has taught the analysis courses since he started teaching both at Clemson and at his previous post at Michigan Technological University.

In between, he spent time as a senior engineer in various aerospace firms and even did a short stint in a software development company. The problems he was exposed to were very hard, and not amenable to solution using just one approach. Using tools from many branches of mathematics, from many types of computational languages, and from first-principles analysis of natural phenomena was absolutely essential to make progress.

In both mathematical and applied areas, students often need to use advanced mathematics tools they have not learned properly. So, he has recently written a series of five books on mathematical analysis to help researchers with the problem of learning new things after they have earned their degrees and are practicing scientists. Along the way, he has also written papers in immunology, cognitive science, and neural network technology, in addition to having grants from the NSF, NASA, and the US Army.

He also likes to paint, build furniture, and write stories.

Author(s): James K. Peterson
Edition: 1
Publisher: Chapman and Hall/CRC
Year: 2021

Language: English
Pages: 574
Tags: Functional Analysis; Topology; Basic Analysis; Functional Linear Analysis; Functional Nonlinear Analysis;

Cover
Half Title
Title Page
Copyright Page
Acknowledgments
Table of Contents
I. Introduction
1. Introduction
1.1. Table of Contents
1.2. Acknowledgments
II. Some Algebraic Topology
2. Basic Metric Space Topology
2.1. Open Sets of Real Numbers
2.2. Metric Space Theory
2.2.1. Open and Closed Sets
2.3. Analysis Concepts in Metric Spaces
2.4. Some Deeper Metric Space Results
2.5. Deeper Vector Space and Set Results
3. Forms and Curves
3.1. When Is a 1-Form Exact?
3.2. Forms on More Complicated Sets
3.3. Angle Functions and Winding Numbers
3.4. A More General Definition of Winding Number
3.5. Homotopies
4. The Jordan Curve Theorem
4.1. Winding Numbers and Topology
4.2. Some Fundamental Results
4.3. Some Applications
4.3.1. The Fundamental Theorem of Algebra
4.4. The Brouwer Fixed Point Theorem
4.5. De Rham Groups and 1-Forms
4.6. The Coboundary Map
4.7. The Inside and Outside of a Curve
III. Deeper Topological Ideas
5. Vector Spaces and Topology
5.1. Topologies and Topological Spaces
5.1.1. Topological Generalizations of Analysis Concepts
5.1.2. Urysohn's Lemma
5.2. Constructing Topologies from Simpler Sets
5.3. Urysohn's Metrization Theorem
5.4. Topological Vector Spaces
5.4.1. Separation Properties of Topological Vector Spaces
6. Locally Convex Spaces and Seminorms
6.1. Additional Classifications of Topological Vector Spaces
6.1.1. Local Convexity Results
6.2. Metrization in a Topological Vector Space
6.3. Constructing Topologies
6.4. Families of Seminorms
6.5. Another Metrization Result
6.6. A Topology for Test Functions
6.6.1. The Test Functions as a Topological Vector Space
6.6.2. Properties of the Topological Vector Space D(R)
7. A New Look at Linear Functionals
7.1. The Basics
7.2. Locally Convex Topology Examples
7.2.1. A Locally Convex Topology on Continuous Functions
7.2.2. A Locally Convex Topology on All Sequences
7.3. Total Sets and Weak Convergence
8. Deeper Results on Linear Functionals
8.1. Closed Operators and Normed Linear Spaces
8.2. Closed Operators and Topological Spaces
8.3. Extensions to Metric Linear Spaces
8.4. Linear Functional Results
8.5. Early Banach - Alaoglu Results
8.6. The Full Banach - Alaoglu Result
8.7. Separation Ideas
8.8. Krein - Milman Results
9. Stone - Weierstrass Results
9.1. Weierstrass Approximation Theorem
9.2. Partial Orderings
9.3. Continuous Functions on a Topological Space
9.4. The Stone - Weierstrass Theorem
IV. Topological Degree Theory
10. Brouwer Degree Theory
10.1. Construction of n - Dimensional Degree
10.1.1. Defining the Degree of a Mapping
10.1.2. Sard's Theorem
10.2. The Properties of the Degree
10.3. Fixed Point Results
10.4. Borsuk's Theorem
10.5. Further Properties of Brouwer Degree
10.6. Extending Brouwer Degree to Finite Dimensional Normed Linear Spaces
11. Leray - Schauder Degree
11.1. Zeroing in on an Infinite Dimensional Degree
11.2. Properties of Leray - Schauder Degree
11.3. Further Properties of Leray - Schauder Degree
11.4. Linear Compact Operators
11.4.1. The Resolvent Operator
11.4.2. The Spectrum of a Bounded Linear Operator
11.4.3. The Ascent and Descent of an Operator
11.4.4. More on Compact Operators
11.4.5. The Eigenvalues of a Compact Operator
12. Coincidence Degree
12.1. Functional Analysis Background
12.2. The Development of Coincidence Degree
12.2.1. The Generalized Inverse of a Linear Fredholm Operator of Index Zero
12.2.2. Applying Leray - Schauder Degree Tools
12.2.3. The Leray - Schauder Tools Dependence on J, P and Q
12.2.4. The Leray - Schauder Degree for L + G = 0 is Independent of P and Q
12.2.5. The Definition of Coincidence Degree
12.3. Properties of Coincidence Degree
12.4. Further Properties of Coincidence Degree
12.5. The Dependence of Coincidence Degree on Operator Splitting
12.6. Applications of Topological Degree Methods to Boundary Value Problems
V. Manifolds
13. Manifolds
13.1. Manifolds: Definitions and Properties
13.1.1. Implicit Function Manifolds
13.1.2. Projective Space
14. Smooth Functions on Manifolds
14.1. The Tangent Space
14.1.1. Basis Vectors for the Tangent Space
14.1.2. Change of Basis Results
14.2. The Cotangent Space
14.3. The Duality between the Tangent and Cotangent Space
14.4. The Differential of a Map
14.4.1. Tangent Vectors of Curves in a Manifold
14.5. The Tangent Space Using Curves
15. The Global Structure of Manifolds
15.1. Vector Fields and the Tangent Bundle
15.2. The Tangent Bundle
15.3. Vector Fields Revisited
15.4. Tensor Analysis on Manifolds
15.5. Tensor Fields
15.6. Metric Tensors
15.7. The Riemannian Metric
VI. Emerging Topologies
16. Asynchronous Computation
16.1. Gene Viability
16.2. SIR Disease Models
16.3. Associations in Complex Graph Models
16.4. Comments on Feedback Graph Models
16.4.1. Information Flow
16.4.2. A Constructive Example
16.5. Sudden Complex Model Changes Due to an External Signal
16.5.1. Zombie Creation in Anesthesia
16.5.2. Zombie Creation from Parasites
16.5.3. iZombie Creation from an Epileptic Episode
16.5.4. Changes in Cognitive Processing Due to External Drug Injection
16.5.5. What Does This Mean?
16.5.6. Lesion Studies
16.6 Message Passing Architectures
16.6.1. Computational Graph Models for Information Processing
16.6.2. Asynchronous Graph Models
16.6.3. Breaking the Initial Symmetry
16.6.4. Message Sequences
16.6.5. Breaking the Symmetry Again: Version One
16.6.6. Breaking the Symmetry Again: Version Two
16.6.7. Topological Considerations
16.6.8. The Signal Network
16.6.9. Asynchronous Computational Graph Models
17. Signal Models and Autoimmune Disease
17.1. Antigen Pathway Models
17.2. Two Allele LRRK2 Mutation Models
17.3. Signaling Models
17.4. The Avidity Calculation
17.5. A Simple Cytokine Signaling Model
17.6. Sample Self-Damage Scenarios
17.7. The Asynchronous Graph Neurodegeneration Model
18. Bar Code Computations in Consciousness Models
18.1. General Graph Models for Information Processing
18.2. The Asynchronous Immune Graph Model
18.3. The Cortex-Thalamus Computational Loop
18.4. Cortical Representation and Cognitive Models
18.5. What Does This Mean?
VII. Summing It All Up
19. Summing It All Up
VIII. References
IX. Detailed Index