Basic Analysis V: Functional Analysis and Topology
- Length: 574 pages
- Edition: 1
- Language: English
- Publisher: Chapman and Hall/CRC
- Publication Date: 2021-09-13
- ISBN-10: 1138055131
- ISBN-13: 9781138055131
- Sales Rank: #0 (See Top 100 Books)
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.
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
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