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

\def\C{{\mathbb C}}
\def\e{{\mathrm e}}
\def\d{{\mathrm d}}
\def\interior{\operatorname{int}}
\def\N{{\mathbb N}}
\def\R{{\mathbb R}}

\begin{document}

\title{Functional Analysis}
\author{Homework 2}
\date{due February 20, 2009\\(submit by 17:00 in Research I, Room 40)}
\maketitle

\begin{enumerate}

\item (From Folland, p.\ 124.) Let $X$ and $Y$ be topological spaces.
\begin{enumerate}
\item If $X$ is connected (see Homework 1) and $f \in C(X,Y)$,
then $f(X)$ is connected.
\item $X$ is called \emph{pathwise connected} (or \emph{arcwise
connected}) if for all $x_0,x_1 \in X$ there exists $f \in C([0,1],X)$
with $f(0)=x_0$ and $f(1)=x_1$.  Show that every pathwise connected
space is connected.
\item Let $X=\{(s,t) \in \R^2 \colon t = \sin(1/s)\} \cup \{(0,0)\}$,
with the relative topology induced from $\R^2$.  Then $X$ is
connected, but not pathwise connected.
\end{enumerate}

\item (From Folland, p.\ 138.)  If $\{X_\alpha\}_{\alpha \in A}$ is a
family of topological spaces of which infinitely many are noncompact,
then every closed compact subset of $\prod_{\alpha \in A} X_\alpha$ is
nowhere dense.

(Recall that a subset $S$ of a topological space is called
\emph{nowhere dense} if $\interior (\overline S) = \emptyset$.)

\item (From Folland, p.\ 138.)  Let $K \in C([0,1]^2)$.  For $f \in
C([0,1])$, let
\[
  Tf(x) = \int_0^1 K(x,y) \, f(y) \, \d y \,.
\]
Then $Tf \in C([0,1])$, and
\[
  \{ Tf \colon \lVert f \rVert_u \leq 1 \}
\]
is precompact in $C([0,1])$.

(A subset $S$ of a topological space is called \emph{precompact} (or
\emph{relatively compact}) if its closure is compact.)

\item (From Folland, p.\ 138.)  Let $(X,\rho)$ be a metric space.  A
function $f \in C(X)$ is called \emph{H\"older continuous of exponent}
$\alpha>0$ if the quantity
\[
  N_\alpha(f) = \sup_{x\neq y}
  \frac{\lvert f(x)-f(y) \rvert}{\rho(x,y)^\alpha}
\]
is finite.  If $X$ is compact, prove that
\[
  \{ f \in C(X) \colon \lVert f \rVert_u \leq 1 \text{ and }
     N_\alpha(f) \leq 1 \}
\]
is compact in $C(X)$.


\end{enumerate}

\end{document}
