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\begin{document}
\begin{center}
{\huge \textbf{Syllabus for Oral Examination}}\\[.35cm]
{\Large \textbf{Haoyuan Xu}}\\[0.5cm]
{\LARGE \textbf{Functional Analysis}}\\[0.35cm]
\end{center}
\flushleft
{\large \textbf{Banach spaces}}\\[0.3cm]
\begin{enumerate}
\item Linear Normed Spaces
\item Basis and Barie Category Theorem
\item Strong, Weak and Weak* Convergence
\item Reflexibility of a Banach Space
\item Dual Spaces
\item Hahn-Banach Theorem
\item Alaogulu Theorem
\end{enumerate}
\vspace{0.5cm}
{\Large \textbf{Hilbert Spaces}}\\[0.3cm]
\begin{enumerate}
\item Riezs Lemma
\item Orthonormal sets and Bases
\item Bessel's inequality and arseval's Theorem
\end{enumerate}
\vspace{0.5cm}
{\Large \textbf{Linear Operators}}
\vspace{0.5cm}
\begin{enumerate}
\item Uniform Bounded Principle
\item Open Mapping Theorem, Inverse Mapping Theorem and Closed Graph Theorem
\item Fixed Points Theorem
\item Riezs-Thorin Interpolation Theroem
\item Compact Operators
\begin{itemize}
\item Resolvent Set
\item Resolvent Operators
\item Spectrum
\end{itemize}
\item Spectrum of a Compact Operator
\item Self-adjoint Operator
\end{enumerate}
\vspace{0.5cm}


\begin{center}
{\LARGE \textbf{Partial Differential Equations}}\\[0.5cm]
\end{center}
\flushleft
{\Large \textbf{Laplacian Equation}}\\[0.3cm]
\begin{enumerate}
\item Mean-value Properties
\item The Maximum Principle
\item Harnack Inequality
\item Perron's Method
\item Green's Representation
\item Fundamental Solution
\item Newtonian Potential
\end{enumerate}
\vspace{0.5cm}

{\large \textbf{Sobolev Spaces}}\\[0.3cm]
\begin{enumerate}
\item Extension Theorem
\item Sobolev Embedding Theorem
\item Sobolev Inequalities(Morrey,Gagliardo-Nirenberg-Sobolev Inequality)
\item Poicar\'e's Inequality
\item Compact Imbedding Theorem
\end{enumerate}
\vspace{0.5cm}
{\Large \textbf{Second Order Elliptic Equations}}\\[0.3cm]
\begin{enumerate}
\item Classical Maximum Principle
\item Uniqueness of Dirichlet and Neumann Boundary Value Problems
\item Existence theorems
\begin{itemize}
\item Method of Continuity
\item Lax-Milgram Theorem
\item Fredholm Alternative Theorem
\end{itemize}
\item Regularity of Solutions and Schauder Interior and Global Estimates
\end{enumerate}
\end{document}
