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\title{Oral Exam Syllabus}
\author{Brian Lins}
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\date{Committee: 
Roger Nussbaum, Sagun Chanillo, Zheng-Chao Han and Richard Wheeden}



\section{Functional Analysis}

\subsection{Topological Vector Spaces}
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~~A. Basic Properties and Examples

~~B. Locally Convex Spaces

~~C. Banach Spaces

\subsection{Baire Category Theorem and Consequences}
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~~A. Baire Category

~~B. Uniform Boundedness Principle

~~C. Open Mapping Theorem

~~D. Closed Graph Theorem

\subsection{Duality and Convexity}
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~~A. Hahn-Banach Theorems

~~B. Banach-Alaoglu Theorem

~~C. Krein-Milman Theorem

\subsection{Fixed Point Theorems and Applications}
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~~A. Basic Fixed Point Theorems (Brouwer and Contraction Principle)

~~B. Markov-Kakutani

~~C. Local Solutions of ODE

\subsection{Hilbert Spaces}
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~~A. Geometry of Hilbert Space

~~B. The Algebra of Operators

~~C. Spectral Theorem for Hermitian and Normal Operators

~~D. Unbounded Operators

\section{Harmonic Analysis}

\subsection{Fourier Transform}
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~~A. Construction and Basic Properties

~~B. Application to $L^2$

~~C. Schwartz Space

~~D. Tempered Distributions

~~E. Operators that Commute with Translations

\subsection{Some Fundamental Notions}
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~~A. The Hardy-Littlewood Maximal Function

~~B. Lebesque Differentiation Theorem

~~C. Vitali Covering Lemma

~~D. Marcinkiewicz Integral

~~E. Calderon-Zygmund Decomposition

\subsection{Interpolation of Operators}
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~~A.  Riesz-Thorin

~~B.  Marcinkiewicz Interpolation Theorem

\subsection{Singular Integrals}
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~~A. $L^P$ Boundedness

~~B. Hilbert Transform

~~C. Riesz Transform

~~D. Applications

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Primary References: 

~~~~Rudin, \emph{Functional Analysis}

~~~~Stein, \emph{Singular Integrals and Differentiability Properties of Functions}

~~~~Stein and Weiss, \emph{Introduction to Fourier Analysis on Euclidean Spaces}


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