\documentclass[11pt]{article}
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{epstopdf}
\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png}

\textwidth = 6.5 in
\textheight = 9 in
\oddsidemargin = 0.0 in
\evensidemargin = 0.0 in
\topmargin = 0.0 in
\headheight = 0.0 in
\headsep = 0.0 in
\parskip = 0.2in
\parindent = 0.0in

\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}{Definition}

\begin{document}

\begincenter{}\endcenter






\begin{center}
  {\bf Basic Real Analysis, Math 603-4}

{\bf Fall 98 - Spring 99}

{\it A. Alexandrou Himonas}
\end{center}

 Real Analysis has its roots in the work of Archimedes(287-212 BC) 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
analysis were understood only in the Twentieth century starting with the
development of a new integral by Lebesgue (1900-1950).
Today Real Analysis is still a growing subject in Mathematics
with applications to problems from Social Sciences,
Business and Economics to Engineering and the Physical
Sciences. Our 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.

The central theme of the first semester will be
the Lebesgue measure and integral,
together with the various notions of convergence of a sequence of
functions.  In addition we will discuss abstract measures and integrals,
being  generalizations of the Lebesgue measure and
integral.  We will start with a review of the notions of  limit,
continuity, compactness, and the Riemann integral in the n-dimensional
Euclidean space.  And, we will end with differentiation of measures.


In the second semester we will start with an introduction to Banach and
Hilbert spaces, and will continue with $L^p$ spaces and Radon measures.
In the last part of the semester we will try to present the following topics:
An introduction to Fourier analysis,
Sobolev spaces and applications to partial differential
equations, an introduction to probability and the Brownian motion,
and applications to mathematical economics.

\vskip .25 in
\centerline{\bf Textbooks}
\noindent
1. {\bf G.B. Folland.}  Real Analysis, Modern Techniques
and Applications, Willey, ISBN 0-471-80958-6

\noindent
2. {\bf R.L. Wheeden and A. Zygmund.}  Measure and
Integral, an  Introduction to Real Analysis,  Marcel Dekker, ISBN
0-8247-6499-4.


\vskip .25 in
\centerline{\bf References}
 \noindent
1. {\bf M. Reed and B. Simon.}  Methods of Modern Mathematical Physics, I:
Functional Analysis, Academic Press, 2nd edition, ISBN 0-12-585001-8

\noindent
2. {\bf E. Evans and R.F. Gariepy.}  Measure Theory and Fine Properties
of Functions,  CRC Press, ISBN 0-8493-7157-0.

\noindent
3. {\bf G.B. Folland.} Introduction to Partial Differential Equations,
Princeton University Press, 2nd Edition, ISBN 0-691-04361-2.

\noindent
4. {\bf F. John.} Partial Differential Equations,
Springer-Verlag, 4th Edition, ISBN 0-387-90609-6.

\noindent
5. {\bf P. Malliavin.}  Integration and Probability,
Springer-Verlag, ISBN 0-387-94409-5.

\noindent
6. {\bf D.W. Strook.}   Probability Theory, an Analytic View,
Cambridge University Press, ISBN 0-521-43123-9.

\noindent
7. {\bf R. Durrett.}   Stochastic Calculus,
CRC Press, ISBN 0-8493-8071-5.

\noindent
8. {\bf N.L. Stokey and R.E. Lucas, Jr.}  Recursive Methods in Economic
Dynamics, Harvard University Press,
ISBN 0-647-75096-9



\begin{center}
  {\bf  Syllabus}
\end{center}


\begin{description}
\item[1.] Preliminaries\hspace{1truecm}
  \begin{description}
  \item[1. ] Real Numbers
  \item[2. ] Metric Spaces
  \item[3. ] Compact sets; The Heine-Borel Theorem
  \item[4. ] Continuity
  \item[5. ] Contraction Mapping Theorem
  \item[6. ] The Basic Theorem for Existence and Uniqueness of Solutions
             to Ordinary Differential Equations
  \item[7. ] The Ascoli-Arzel\' a Theorem
  \item[8. ] The Weierstrass Approximation Theorem
  \item[9. ] The Baire Category Theorem
  \item[10.] Review of he Riemann Integral

   \end{description}
\item[2.] Measures
  \begin{description}
  \item[1. ] $\sigma$-algebras
  \item[2. ] Measures
  \item[3. ] Outer Measures
  \item[4. ] Borel Measures on $\Bbb R$
  \item[5. ] Lebesgue Measure on $\Bbb R$
 \end{description}

\item[3.] Integration.
  \begin{description}
  \item[1. ] Measurable Functions.
  \item[2. ] Integration of Non-negative Functions
  \item[3. ] Integration of Complex Functions
  \item[4. ] Modes of Convergence
  \item[5. ] Product Measures
  \item[6. ] The Lebesgue Integral on $\Bbb R^n$
 \end{description}
\item[4.] Decomposition and Differentiation of Measures
  \begin{description}
  \item[1. ] Signed Measures
  \item[2. ] The Lebesgue-Radon-Nikodym Theorem
  \item[3. ] Complex Measures
  \item[4. ] Differentiation in $\Bbb R^n$
  \item[5. ] Functions of Bounded Variation
 \end{description}

 \item[5.] Introduction to Functional Analysis
  \begin{description}
  \item[1. ] Normed Vector Spaces
  \item[2. ] Linear Functionals; Hahn-Banach Theorem
  \item[3. ] The Open Mapping Theorem
  \item[4. ] The Closed Graph Theorem
  \item[5. ] The Uniform Boundedness Principle
  \item[6. ] Hilbert Spaces, Pythagorian Theorem, and Orthonormal Bases
  \item[7. ] Spectral Theory for Compact Self-adjoint Operators on a
              Hilbert space
 \end{description}
\item[6.] $ L^p$ Spaces
  \begin{description}
  \item[1. ] Basic Theory, Holder's and Minkowski's Inequality
  \item[2. ] The Dual of $L^p$
  \item[3. ] Some Useful Inequalities
  \item[4. ] Interpolation of $L^p$ spaces
  \end{description}
\item[7.] Radon Measures
  \begin{description}
  \item[1. ] Positive Linear functionals on $C_c(X)$ and  the Riesz
Representation Theorem
  \end{description}
\item[8.] Topics in Fourier Analysis
  \begin{description}
  \item[1. ] Convolutions
  \item[2. ] The Fourier Transform
  \item[3. ] The Fourier Inversion Theorem
  \item[4. ] Fourier Series and Convergence Theorems
  \item[5. ] The Fourier Transform and Smoothness of Functions
  \end{description}
\item[9.] Distributions and Sobolev Spaces
  \begin{description}
  \item[1. ] Distributions and Weak Derivatives
  \item[2. ] Sobolev Spaces, the Sobolev Embedding Theorem
  \item[3. ] Applications to Partial Differential Equations
   \end{description}
\item[10.] Topics in Probability Theory
  \begin{description}
  \item[1. ] Basic Concepts
  \item[2. ] The Law of Large Numbers
  \item[3. ] The Central Limit Theorem
  \item[4. ] Stochastic Processes and  Construction of the Brownian Motion
(Wiener Process)
  \end{description}
\item[11.] Applications to Mathematical Economics
\end{description}

\end{document}


�����ܥe-�	e�I/�.j.jj.j.j.j.j.~.~.~.~.~.~.�."~.�.?�.�.�.�.�.�.�.�.�.�.�.�.�.�.�.�.TI/�.j.�.�.�.�.�.�.�.j.j.�.�.�.�.�.�.j.�.j.�.�.~.~.j.j.j.j.�.�.�.�.\documentclass{article}
\usepackage{amssymb}
\renewcommand{\baselinestretch}{1.4}

\begin{document}
\begin{center}
  {\bf Basic Real Analysis, Math 603-4}

{\bf Fall 98 - Spring 99}

{\it A. Alexandrou Himonas}
\end{center}

 Real Analysis has its roots in the work of Archimedes(287-212 BC) 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
analysis were understood only in the Twentieth century starting with the
development of a new integral by Lebesgue (1900-1950).
Today Real Analysis is still a growing subject in Mathematics
with applications to problems from Social Sciences,
Business and Economics to Engineering and the Physical
Sciences. Our 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.

The central theme of the first semester will be
the Lebesgue measure and integral,
together with the various notions of convergence of a sequence of
functions.  In addition we will discuss abstract measures and integrals,
being  generalizations of the Lebesgue measure and
integral.  We will start with a review of the notions of  limit,
continuity, compactness, and the Riemann integral in the n-dimensional
Euclidean space.  And, we will end with differentiation of measures.


In the second semester we will start with an introduction to Banach and
Hilbert spaces, and will continue with $L^p$ spaces and Radon measures.
In the last part of the semester we will try to present the following topics:
An introduction to Fourier analysis,
Sobolev spaces and applications to partial differential
equations, an introduction to probability and the Brownian motion,
and applications to mathematical economics.

\vskip .25 in
\centerline{\bf Textbooks}
\noindent
1. {\bf G.B. Folland.}  Real Analysis, Modern Techniques
and Applications, Willey, ISBN 0-471-80958-6

\noindent
2. {\bf R.L. Wheeden and A. Zygmund.}  Measure and
Integral, an  Introduction to Real Analysis,  Marcel Dekker, ISBN
0-8247-6499-4.


\vskip .25 in
\centerline{\bf References}
 \noindent
1. {\bf M. Reed and B. Simon.}  Methods of Modern Mathematical Physics, I:
Functional Analysis, Academic Press, 2nd edition, ISBN 0-12-585001-8

\noindent
2. {\bf E. Evans and R.F. Gariepy.}  Measure Theory and Fine Properties
of Functions,  CRC Press, ISBN 0-8493-7157-0.

\noindent
3. {\bf G.B. Folland.} Introduction to Partial Differential Equations,
Princeton University Press, 2nd Edition, ISBN 0-691-04361-2.

\noindent
4. {\bf F. John.} Partial Differential Equations,
Springer-Verlag, 4th Edition, ISBN 0-387-90609-6.

\noindent
5. {\bf P. Malliavin.}  Integration and Probability,
Springer-Verlag, ISBN 0-387-94409-5.

\noindent
6. {\bf D.W. Strook.}   Probability Theory, an Analytic View,
Cambridge University Press, ISBN 0-521-43123-9.

\noindent
7. {\bf R. Durrett.}   Stochastic Calculus,
CRC Press, ISBN 0-8493-8071-5.

\noindent
8. {\bf N.L. Stokey and R.E. Lucas, Jr.}  Recursive Methods in Economic
Dynamics, Harvard University Press,
ISBN 0-647-75096-9



\begin{center}
  {\bf  Syllabus}
\end{center}


\begin{description}
\item[1.] Preliminaries\hspace{1truecm}
  \begin{description}
  \item[1. ] Real Numbers
  \item[2. ] Metric Spaces
  \item[3. ] Compact sets; The Heine-Borel Theorem
  \item[4. ] Continuity
  \item[5. ] Contraction Mapping Theorem
  \item[6. ] The Basic Theorem for Existence and Uniqueness of Solutions
             to Ordinary Differential Equations
  \item[7. ] The Ascoli-Arzel\' a Theorem
  \item[8. ] The Weierstrass Approximation Theorem
  \item[9. ] The Baire Category Theorem
  \item[10.] Review of he Riemann Integral

   \end{description}
\item[2.] Measures
  \begin{description}
  \item[1. ] $\sigma$-algebras
  \item[2. ] Measures
  \item[3. ] Outer Measures
  \item[4. ] Borel Measures on $\Bbb R$
  \item[5. ] Lebesgue Measure on $\Bbb R$
 \end{description}

\item[3.] Integration.
  \begin{description}
  \item[1. ] Measurable Functions.
  \item[2. ] Integration of Non-negative Functions
  \item[3. ] Integration of Complex Functions
  \item[4. ] Modes of Convergence
  \item[5. ] Product Measures
  \item[6. ] The Lebesgue Integral on $\Bbb R^n$
 \end{description}
\item[4.] Decomposition and Differentiation of Measures
  \begin{description}
  \item[1. ] Signed Measures
  \item[2. ] The Lebesgue-Radon-Nikodym Theorem
  \item[3. ] Complex Measures
  \item[4. ] Differentiation in $\Bbb R^n$
  \item[5. ] Functions of Bounded Variation
 \end{description}

 \item[5.] Introduction to Functional Analysis
  \begin{description}
  \item[1. ] Normed Vector Spaces
  \item[2. ] Linear Functionals; Hahn-Banach Theorem
  \item[3. ] The Open Mapping Theorem
  \item[4. ] The Closed Graph Theorem
  \item[5. ] The Uniform Boundedness Principle
  \item[6. ] Hilbert Spaces, Pythagorian Theorem, and Orthonormal Bases
  \item[7. ] Spectral Theory for Compact Self-adjoint Operators on a
              Hilbert space
 \end{description}
\item[6.] $ L^p$ Spaces
  \begin{description}
  \item[1. ] Basic Theory, Holder's and Minkowski's Inequality
  \item[2. ] The Dual of $L^p$
  \item[3. ] Some Useful Inequalities
  \item[4. ] Interpolation of $L^p$ spaces
  \end{description}
\item[7.] Radon Measures
  \begin{description}
  \item[1. ] Positive Linear functionals on $C_c(X)$ and  the Riesz
Representation Theorem
  \end{description}
\item[8.] Topics in Fourier Analysis
  \begin{description}
  \item[1. ] Convolutions
  \item[2. ] The Fourier Transform
  \item[3. ] The Fourier Inversion Theorem
  \item[4. ] Fourier Series and Convergence Theorems
  \item[5. ] The Fourier Transform and Smoothness of Functions
  \end{description}
\item[9.] Distributions and Sobolev Spaces
  \begin{description}
  \item[1. ] Distributions and Weak Derivatives
  \item[2. ] Sobolev Spaces, the Sobolev Embedding Theorem
  \item[3. ] Applications to Partial Differential Equations
   \end{description}
\item[10.] Topics in Probability Theory
  \begin{description}
  \item[1. ] Basic Concepts
  \item[2. ] The Law of Large Numbers
  \item[3. ] The Central Limit Theorem
  \item[4. ] Stochastic Processes and  Construction of the Brownian Motion
(Wiener Process)
  \end{description}
\item[11.] Applications to Mathematical Economics
\end{description}

\end{document}
�������