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.
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