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{\bf Math 604, Real Analysis, Spring, 2004}
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\noindent {\bf Instructor:} Qing Han

228 Hayes-Healy, 631-6834

email: qhan@nd.edu

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\noindent {\bf Text book}:  {\it Real Analysis} by Folland.

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Real analysis is the foundation on which many theory are built.
Math 604 is a continuation of Math 603. In Math 604, the following
materials will be covered.


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1. Banach spaces. Dual-spaces and their conjugates, the
Riesz-Fisher Theorem, the Riesz Representation Theorem for bounded
linear functionals on $C(X)$, the Riesz Representation Theorem for
$C(X)$, the Hahn-Banach Theorem, the Closed Graph and Open Mapping
Theorems, the Principle of Uniform Boundedness, Alaoglu's Theorem,
Hilbert spaces, orthogonal systems, Fourier series, Bessel's
inequality, Parseval's formula, convolutions, Fourier transform,
distributions.

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2. Sobolev spaces. Marcinkiewicz interpolation theorem,
Calderon-Zygmund decomposition lemma, singular integrals and $L^p$
estimates for Newtonian potential.

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3. Families of functions. Equicontinuous families and the
Arzela-Ascoli Theorem, the Stone-Weierstrass Theorem.


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\noindent{\bf Exams:} There will be a mid-term exam and a final
exam. Both are take-home exams.

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\noindent{\bf Grade:} The course grade is based on weekly homework
and exams.
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