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\title{Functional Analysis}
\author{Final Exam}
\date{May 20, 2009}
\maketitle

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\begin{enumerate}

\item 
\begin{enumerate}
\item Let $E$ be a Banach space with closed subspace $G \subsetneq
E$.  Show that $G$ is nowhere dense in $E$.
\item Let $V$ be a vector space with a countably infinite (algebraic)
basis $B$, i.e.\ every $x \in V$ can be represented as a finite linear
linear combination of elements from $V$.  Show that $E$ cannot be
complete with respect to \emph{any} norm.
\end{enumerate}

\item Let $E$ be a Banach space and $A \in \L(E)$.  Show that
$\sigma(A)$ is compact.

\item Let $H$ be a Hilbert space and $A$ a linear, closed, densely
defined, skew-adjoint (i.e.\ $A^*=-A$), unbounded operator on $H$.
Show that 
\begin{enumerate}
\item $\sigma(A) \subset \i \R$;
\item $A$ has no residual spectrum.
\end{enumerate}

\item Let $H = \ell^2(\N)$ and define an operator $A$ via $A e_n =
n^{-1} \, e_n$ for $n \in \N$, where $\{e_n\}$ is the canonical basis
in $\ell^2$.  Further, let $L$ be the left shift operator on $\ell^2$,
i.e.\ $L(a_1, a_2, a_3, \dots) = (a_2, a_3, \dots)$.
\begin{enumerate}
\item Show that $LA$ is compact.
\item What is $\sigma(LA)$?
\end{enumerate}

\item Let $E,F$ be Banach spaces and $A \colon \D(A) \subset E \to F$
be a linear, closed, densely defined, and invertible unbounded
operator.  Show that $A^*$ is invertible with inverse $(A^{-1})^*$.

\item Consider $A=\partial_x$ on $L^2([0,1])$ with
\[
  \D(A) = \{ u \in L^2 \colon \partial_x u \in L^2([0,1]), u(0)=0 \}
  \,. 
\]
\begin{enumerate}
\item
Show that $A^* = -\partial_x$ with
\[
  \D(A^*) = \{ u \in L^2 \colon \partial_x u \in L^2([0,1]), u(1)=0 \}
  \,. 
\]
\item
Describe the point, continuous, and residual spectrum.
\end{enumerate}
\end{enumerate}

\end{document}
