2024-03-29T07:49:13Z
http://digitalcommons.conncoll.edu/do/oai/
oai:digitalcommons.conncoll.edu:mathhp-1000
2010-06-21T19:34:45Z
publication:honors
publication:mathematics
publication:mathhp
Functional Analysis and its Applications
Luthy, Peter M.
Honors Paper
2005-04-01T08:00:00Z
functional analysis
mathematics
Ames Prize
<p>Functional analysis is, in short, the study of vector spaces with arbitrary dimension. Developed
in the early twentieth century, it is an extension and amalgamation of the fields of complex analysis
and linear algebra. David Hilbert, Frigyes and Marcel Riesz, and Stefan Banach were some of
the noteworthy mathematicians who pushed the early frontiers of the subject. Many of the ideas
produced from their seminal work led to powerful ideas and applications in related subjects: operator
theory and ideas of Hilbert spaces were applied to physics in the early part of the twentieth century,
and are now of fundamental importance in the field of quantum mechanics, and the theory of
partial differential equations would be nearly impossible without the aid of very general fixed-point
theorems.</p>
<p>The purpose of this paper is threefold. First, we wish to make clear to the reader the structural
similarities and differences between finite- and infinite-dimensional vector spaces. In doing so, the
reader with some knowledge of linear algebra and real or complex analysis should be comfortable
with the generalizations and abstractions contained below. Second, we wish to explore in detail
the mathematical ideas and techniques at the foundation of functional analysis. In other words,
we shall develop a context of study and, in doing so, convey to the reader some sense of what one
actually means by functional analysis. The proofs are often elegant, and the techniques used to
solve problems are quite easily applied to other areas of functional analysis. These two points shall
be the topic of the first three chapters. Third, we wish to impress upon the reader the power of
functional analysis to solve pure and applied problems in other areas of mathematics. This will be
done especially in Chapter 4, where we apply a very general fixed point theorem to a particular
partial differential equation to establish existence and smoothness of generalized solutions. This
latter work was completed during the author's NSF-funded internship at Cornell University and
credit should be also attributed to Phillip Whitman, Frances Hammock, and Alexander Meadows.</p>
https://digitalcommons.conncoll.edu/mathhp/1
oai:digitalcommons.conncoll.edu:mathhp-1001
2017-08-02T18:28:31Z
publication:mathematics
publication:mathhp
Functional Equations and their Applications
Lalov, Emil
Restricted
2014-01-01T08:00:00Z
2014-05-20T07:00:00Z
Mathematics
https://digitalcommons.conncoll.edu/mathhp/2