\documentclass[12pt]{article}
\usepackage{amsmath,amsfonts,fullpage}

\def\C{{\mathbb C}}
\def\e{{\mathrm e}}
\def\d{{\mathrm d}}
\def\interior{\operatorname{int}}
\def\dist{\operatorname{dist}}
\def\Ker{\operatorname{Ker}}
\def\Range{\operatorname{Range}}
\def\N{{\mathbb N}}
\def\R{{\mathbb R}}
\def\T{{\mathbb T}}
\def\L{{\mathcal L}}
\def\K{{\mathcal K}}
\def\D{{\mathcal D}}

\begin{document}

\title{Functional Analysis}
\author{Homework 7}
\date{due April 24, 2009}
\maketitle

\begin{enumerate}

\item Show that a closed subset $A$ of a complete metric space $X$ is
compact if and only if for every $\varepsilon >0$ there exist finitely
many $x_1, \dots, x_k \in X$ such that
\[
  A \subset \bigcup_{i=1}^k B(x_i, \varepsilon) \,.
\]

\item Show that the composition (in any order) of a compact and a
bounded operator is compact.

\item Let $E, F$ be Banach spaces and $T \in \L(E, F)$.  Prove the
following.
\begin{enumerate}
\item If $T$ is compact and $u_n \rightharpoonup u$ weakly in $E$,
then $Tu_n \to Tu$ strongly in $F$.

\item If $E$ is reflexive and $u_n \rightharpoonup u$ weakly in $E$
implies that $Tu_n \to Tu$ strongly in $F$, then $T$ is compact.
\end{enumerate}

\item Show that $T \in \K(E,F)$ if and only if $T^* \in \K(F^*,E^*)$. 

\emph{Note:} The proof is essentially given on p.~76 in the lecture
notes.  Write out a \emph{self-contained} version filling in all the
gaps.  In particular, verify that the assumptions of the
Arzel\`a-Ascoli theorem are satisfied, that $\D(T*) = F*$, and explain
the items marked ``why?''. 

\end{enumerate}
\end{document}
