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\def\classtitle{Real Analysis}
\def\classnumber{Math 70100}
\def\examtitle{Homework 6}
\def\duedate{October 18, 2019}

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\begin{document}
\noindent
{\large \bf \classnumber: \classtitle (Prof. Hooper) \hfill Due \duedate}\\
{\large \bf \examtitle}\\

\ifprintanswers 
\noindent
{\bf Solutions:}
\else
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%{\bf Instructions:}
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\begin{questions}
\question {\em (Folland \S 4.6 \# 49a) }
Show that if $X$ is a compact Hausdorff space and $U \subset X$ is open, then $U$ is locally compact in the relative topology.

\begin{sol}
\end{sol}

\question {\em (Folland \S 4.6 \# 50) }
Let $U$ be an open subset of a compact Hausdorff space $X$ and let $U^\ast$ be the one-point
compactification of $U$ (see previous exercise). Define $\phi : X \to U^\ast$ by $\phi( x) = x$ if $x \in U$ and 
and $\phi( x) = \infty$ if $x \not \in U$. Show $\phi$ is continuous.

\begin{sol}
\end{sol}

\question {\em (Kriz and Pultr \S 9.7 \# 8, compare Folland \S 4.6 \#55)}
Prove that any open set in $\R^n$ is $\sigma$-compact.

\begin{sol}
\end{sol}

\question Assume that $f_n:[0,1] \to \R$ is a sequence of differentiable functions
whose derivatives are uniformly bounded. Suppose there is an $x_0 \in [0,1]$ so that
$\{f_n(x_0)~:~n \in \N\}$ is bounded. Prove that $\{f_n\}$ has a subsequence which converges
uniformly to a continuous function on $[0,1]$.

\begin{sol}
\end{sol}


\question {\em (Royden-Fitzpatrick \S 10.1 \# 5)} A function $f:[0, 1] \to \R$ is said to be H\"older continuous of order $\alpha$ provided there
is a constant $C$ for which
$$|f(x)-f(y)| \leq 
C |x - y|^\alpha \quad \text{for all} \quad x,y \in [0, 1].$$
Define the H\"older norm
$$\|f \|_\alpha=\max~\{|f(x)|+\frac{|f(x)-f(y)|}{|x -y|^\alpha} ~:~ \text{$x,y \in[0, 1]$ and $x \neq y$} \}.$$
Show that for $0 < \alpha < 1$, the set of functions for which $\|f \|_\alpha \leq 1$ has
compact closure as a subset of subset of the space of continuous real-valued functions on $[0,1]$ with the uniform norm.

\begin{sol}
\end{sol}

\question Let $X$ be a compact Hausdorff space and $E$ be a Banach space. Suppose $f_n \in C(X,E)$ is a sequence of continuous functions so that the set $\{f_n \in C(X,E)\}$ is equicontinuous. Further suppose that $A \subset X$ is dense and the sequence $f_n(a)$ converges for all $a \in A$. Prove that the sequence $\{f_n\}$ converges uniformly to some $f \in C(X,E)$.

\begin{sol}
\end{sol}

\question {\em (based on Lang III\S 4 \#9)} 
We say a sequence
$\{f_n\}$ of real valued functions on a topological space $X$ is {\em monotone increasing}
if for each $x \in X$ and each $n \in \N$, $f_{n+1}(x) \geq f_n(x)$. Recall $\{f_n\}$ converges to $f$ pointwise if for each $x \in X$, we have $\lim_{n \to \infty} f_n(x)=f(x)$. 

Prove Dini's theorem: \\
If $\{f_n\}$ is a monotone increasing sequence of continuous real-valued functions on a compact set metric space $X$ which converges pointwise to a continuous function $f: X \to \R$, then 
the sequence converges uniformly.
\begin{sol}
\end{sol}

\end{questions}
\end{document}
