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\begin{document}
\begin{center}
{\huge \textbf{Syllabus for Oral Examination}}\\[.35cm]
{\Large \textbf{Richard Mikula}}\\[0.5cm]
{\LARGE \textbf{Linear Functional Analysis}}\\[0.35cm]
\end{center}
\flushleft
{\large \textbf{Banach spaces}}\\[0.3cm]
\begin{enumerate}
\item Bounded Linear Transformation Theorem
\item Hahn-Banach Theorem and its applications
\item Dual space of a Banach space
\item Canonical embedding of a Banach space $X$ into its double dual, and
reflexive spaces
\item Quotient spaces
\item Baire Category Theorem
\item Banach-Steinhaus Theorem
\item Open Mapping Theorem and Inverse Mapping Theorem
\item Closed Graph Theorem
\end{enumerate}
\vspace{0.5cm}
{\Large \textbf{Topology}}\\[0.3cm]
\begin{enumerate}
\item Compactness
\begin{itemize}
\item Bolzano-Weierstrass Theorem
\item Tychonoff's Theorem
\item Banach-Alaoglu Theorem
\end{itemize}
\item Urysohn's Lemma
\item Norm and weak topologies on a Banach Space $X$
\item The weak$^{*}$ topology on $X^{*}$
\item The uniform, strong, and weak operator topologies
\end{enumerate}
\vspace{0.5cm}
{\Large \textbf{Hilbert spaces}}
\vspace{0.5cm}
\begin{enumerate}
\item Pythagorean identity
\item Parallelogram identity
\item Cauchy-Schwartz inequality
\item Polarization identity
\item Continuity of the inner product
\item Orthonormal sets and bases of Hilbert spaces
\item Bessel's inequality and Parseval's Theorem
\item Projection Theorem and direct sum decomposition
\item Riesz Representation Theorem
\item Hilbert adjoint operator
\end{enumerate}
\vspace{0.5cm}

{\Large \textbf{General spectral theory of an operator $T \in 
L(X,X)$}}\\[0.3cm]
\begin{enumerate}
\item Resolvent set, resolvent operator and the spectrum
\item Power series representation of the resolvent operator
\item Spectral Radius Theorem
\item Non-emptiness of the spectrum
\item Spectral Mapping Theorem
\end{enumerate}
\vspace{0.5cm}


{\Large \textbf{Compact operators}}\\[0.3cm]
\begin{enumerate}
\item Operators with finite dimensional domain and range
\item Integral operators
\item Norm limit of compact operators
\item Spectrum of a compact operator
\item The Fredholm Alterative
\end{enumerate}
\vspace{0.5cm}
{\Large \textbf{Bounded self-adjoint linear operators}}\\[0.3cm]
\begin{enumerate}
\item The spectrum of a self-adjoint operator
\item Positive operators
\item Projection operators
\item The spectral representation of a self-adjoint operator
\end{enumerate}
\vspace{0.5cm}

{\Large \textbf{Unbounded operators}}\\[0.3cm]
\begin{enumerate}
\item Hellinger-Toeplitz Theorem
\end{enumerate}
\vspace{0.5cm}

\begin{center}
{\LARGE \textbf{Elliptic Partial Differential Equations}}\\[0.5cm]
\end{center}
\flushleft
{\Large \textbf{Laplace's equation}}\\[0.3cm]
\begin{enumerate}
\item The fundamental solution
\item Solving Poisson's equation in the H\"older continuous case
\item Mean-value inequalities
\item The maximum principle
\item Harnack inequality
\item Green's representation formula and the Green's function for the unit
ball and half space
\item Converse of mean-value formula for continuous functions
\item Regularity of harmonic functions
\item The Dirichlet problem and Perron's method
\end{enumerate}
\vspace{0.5cm}

{\large \textbf{Sobolev spaces}}\\[0.3cm]
\begin{enumerate}
\item Interior and global approximations by smooth functions
\item Extensions
\item Traces
\item Sobolev estimates
\item Gagliardo-Nirenberg-Sobolev Inequality
\item Poicar\'e's Inequality
\item Morrey's Inequality
\item Rellich-Kondrachov Compactness Theorem
\item Difference quotients
\item Rademacher's Theorem
\end{enumerate}
\vspace{0.5cm}
{\Large \textbf{Second order elliptic equations}}\\[0.3cm]
\begin{enumerate}
\item Lax-Milgram Theorem
\item Energy estimates for the bilinear form $B$ associated to a second
order elliptic operator $L$
\item Existence theorems for weak solutions
\item Eigenvalues and eigenfunctions
\item Regularity theorems: interior and boundary regularity
\item Maximum principles
\end{enumerate}
\end{document}


