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\begin{document}

\begin{center}
{\Large \em Math 560 (1998)}\\
\vspace{.3in}
{\Large Topic Review for Midterm 1}\\
\end{center}

\vspace{.4in}

\large
\begin{itemize}
\item[1.] Linear algebra on $\reals^n, A\in \reals^{n\times n}$.
\begin{itemize}
\item[$\bullet$] Eigenvalues, eigenvectors, diagonalization
\item[$\bullet$] Adjoint, self-adjoint, orthogonal diagonalization
\item[$\bullet$] Fredholm alternative
\item[$\bullet$] Least Squares
\item[$\bullet$] Moore-Penrose Inverse
\item[$\bullet$] Singular Value decomposition
\end{itemize}
\item[2.]  Linear algebra on vector spaces
\begin{itemize}
\item[$\bullet$] Change of bases
\item[$\bullet$] Gram-Schmidt orthogonalization
\item[$\bullet$] Similarity transformations
\end{itemize}
\item[3.] Manifolds
\begin{itemize}
\item[$\bullet$] Linear manifolds
\item[$\bullet$] Invariant manifolds for $A\in \reals^{n \times n}$
\item[$\bullet$] Closed linear manifolds
\end{itemize}
\item[3.] Hilbert Spaces
\begin{itemize}
\item[$\bullet$] Approximation in Hilbert spaces
\item[$\bullet$] Complete (orthonormal) sets
\item[$\bullet$] Bessel's inequaility
\item[$\bullet$] Parseval's equality
\item[$\bullet$] Orthogonal complements (closed)
\item[$\bullet$] Projection theorem
\end{itemize}
\item[4.]  Sturm Liouville Theory 
\begin{itemize}
\item[$\bullet$] Regular S-L problems (defn)
\item[$\bullet$] Basic existence of complete set theorem
\item[$\bullet$] Eigenvalue problems and equations
\item[$\bullet$] Separated and nonseparated boundary conditions
\item[$\bullet$] Regular SL -> orthogonality of eigenfunctions
\item[$\bullet$] Regular SL-> real eigenvalues
\end{itemize}
\item[5.]  Operators
\begin{itemize}
\item[$\bullet$] Nullspace, range
\item[$\bullet$] Bounded, continuous
\item[$\bullet$] Adjoint of a bounded operator
\item[$\bullet$] Fredholm alternative for bounded operators
with closed range
\item[$\bullet$] Compact implies bounded
\item[$\bullet$] Compact operator (defn only)
\item[$\bullet$] 
\end{itemize}
\item[6.] Integral Equation
\begin{itemize}
\item[$\bullet$] Fredholm (separable and nonseparable-defns)
\item[$\bullet$] Volterra (defn)
\item[$\bullet$] Solution of Volterra eqns. by Laplace transform
\item[$\bullet$] Solution of separable Fredholm eqns.
\item[$\bullet$] Eigenvalue problems 
\item[$\bullet$] Conversion of $u''(x)=F(x), u(0)=u(1)=0$ to integral equation
\end{itemize}
\item[7.] Won't cover
\begin{itemize}
\item[$\bullet$] Lattice vibrations
\item[$\bullet$] Maximum principle(s)
\item[$\bullet$] Orthogonal polynomials
\item[$\bullet$] Abel's equation
\item[$\bullet$] Conversion of IVP to Volterra equations
\item[$\bullet$] Functionals, Riesz Representation Theorem
\item[$\bullet$] Topology of compact sets and operators
\item[$\bullet$] Hilbert Schmidt kernels
\item[$\bullet$] Spectral theory for compact operators
\item[$\bullet$] Multidimensional fourier series
\item[$\bullet$] Resolvent operators
\end{itemize}
\end{itemize}

\end{document}