This book is the result of a meeting on Topology and Functional Analysis, and is dedicated to Professor Manuel López-Pellicer's mathematical research. Covering topics in descriptive topology and functional analysis, including topological groups and Banach space theory, fuzzy topology, differentiability and renorming, tensor products of Banach spaces and aspects of Cp-theory, this volume is particularly useful to young researchers wanting to learn about the latest developments in these areas.

"Descriptive Topology in Selected Topics of Functional Analysis" is a collection of recent developments in the field of descriptive topology, specifically focused on the classes of infinite-dimensional topological vector spaces that appear in functional analysis. Such spaces include Fréchet spaces, (LF)-spaces and their duals, and the space of continuous real-valued functions C(X) on a completely regular Hausdorff space X, to name a few. These vector spaces appear in functional analysis in distribution theory, differential equations, complex analysis, and various other analytical settings. This monograph provides new insights into the connections between the topological properties of linear function spaces and their applications in functional analysis.

Descriptive topology and functional analysis, with extensive material demonstrating new connections between them, are the subject of the first section of this work. Applications to spaces of continuous functions, topological Abelian groups, linear topological equivalence and to the separable quotient problem are included and are presented as open problems. The second section is devoted to Banach spaces, Banach algebras and operator theory. Each chapter presents a lot of worthwhile and important recent theorems with an abstract discussing the material in the chapter. Each chapter can almost be seen as a survey covering a particular area.

Even the simplest mathematical abstraction of the phenomena of reality the real line-can be regarded from different points of view by different mathematical disciplines. For example, the algebraic approach to the study of the real line involves describing its properties as a set to whose elements we can apply" operations," and obtaining an algebraic model of it on the basis of these properties, without regard for the topological properties. On the other hand, we can focus on the topology of the real line and construct a formal model of it by singling out its" continuity" as a basis for the model. Analysis regards the line, and the functions on it, in the unity of the whole system of their algebraic and topological properties, with the fundamental deductions about them obtained by using the interplay between the algebraic and topological structures. The same picture is observed at higher stages of abstraction. Algebra studies linear spaces, groups, rings, modules, and so on. Topology studies structures of a different kind on arbitrary sets, structures that give mathe matical meaning to the concepts of a limit, continuity, a neighborhood, and so on. Functional analysis takes up topological linear spaces, topological groups, normed rings, modules of representations of topological groups in topological linear spaces, and so on. Thus, the basic object of study in functional analysis consists of objects equipped with compatible algebraic and topological structures.

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.