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\noindent
\centerline{\bf University of Notre Dame}
\centerline{\bf Mathematics Department}
\centerline{\bf MATH 607, Fall 1997}
 \bigskip
\centerline{\bf Alex Himonas}
\bigskip
\noindent


\noindent Real Analysis has its roots in the work of Archimedes and other ancient
Greek mathematicians who developed techniques for finding areas and volumes.  In
the Seventeenth Century these techniques were further developed  by Newton and
Leibniz into the theory of Calculus.  In the Eighteenth and Nineteenth
Centuries, the power of calculus was applied to the study of  many problems of
both practical and theoretical interest.  For example, Fourier (1768-1830) used
calculus (Fourier series) to solve the heat equation.  However, most of the
concepts underlying Fourier analyses were understood in the Twentieth Century
starting with the development of a new integral by Lebesque
(1900-1950).\par\bigskip  

\noindent CONTENT:\par
\noindent The Lebesque measure and integral, together with the various notions of
convergence of a sequence of functions, will form the central theme of the
first semester.  In addition, we shall discuss the relation between integration
and differentiation.  We will start with a review of the notions of the limit,
continuity, compactness, and the Riemann integral in the n-dimensional Euclidian
space.  And, we will end with abstract measures and integrals being a
straightforward generalization of the Lebesque measure and integral.\par\bigskip

\noindent We will begin the second semester with an introduction to Banach and
Hilbert spaces, and will continue with LP spaces and Radon measures.   The last
part of the semester will cover:\par\bigskip

$\bullet$ 	An introduction to Fourier Analysis;\par\bigskip

$\bullet$	Sobolev spaces and applications to partial differential
equations;\par\bigskip

$\bullet$	An introduction to probability and the Brownian motion;\par\bigskip

$\bullet$	Contraction mapping theorem with applications to mathematical economics
and the proof of existence and uniqueness of solutions for a differential
equation;\par\bigskip

$\bullet$	Spectral theory for compact self-adjoint operators on a Hilbert
space.\par\bigskip

\noindent The objective in this course is to present the essentials of modern
analysis together with some of its applications in the study of both practical
and theoretical problems.  Analysis is a live subject, and we shall try to
present it as such.\par\bigskip

\noindent TEXTBOOKS:\par\bigskip
\noindent 1. \underbar{Folland, G. B.},\  {\it Real Analysis, Modern Techniques
and Application},\  Willey, ISBN 0-471-80958-6.\par\bigskip

\noindent 2.	\underbar{Sheeden, R. L.  and A. Zygmund},\ {\it Measure and
Integral, An  Introduction to Real Analysis},\   Marcel Dekker, ISBN
0-8247-6499-4.


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