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\begin{document}
\begin{center}
\begin{large}
Math 70100: Functions of a Real Variable I \\
Homework 7, due Wednesday, October 22.
\end{large}
\end{center}
{\bf Name:} Insert your name here.
\begin{questions}
\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}
Insert answer here.
\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
$$|f(x)-f(y)| \leq
C |x - y|^\alpha \quad \text{for all} x,y \ in [0, 1].$$
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}
Insert answer here.
\end{sol}
\question {\em (Lang \S III.4 \#21)} Let $X$ be a metric space and $E$ be a normed vector space. Let $BC(X,E)$
be the space of bounded continuous maps $X \to E$ (with the uniform norm).
Let $\Phi$ be a bounded subset of $BC(X,E)$. For $x \in X$, let $\text{ev}_x:\Phi \to E$ be
the function $\text{ev}_x(\phi)=\phi(x)$. Show that $\text{ev}_x$ is continuous and bounded.
Show that $\Phi$ is equicontinuous at a point $a \in X$ if and only if the map $x \mapsto \text{ev}_x$ of $X$ into $BC(\Phi,E)$
is continuous at $a$.
\begin{sol}
Insert answer here.
\end{sol}
\question {\em (Rudin's Principles of real analysis, Chapter 7 \# 20)}
Prove that if $f:[0,1] \to \R$ is continuous and if
$$\int_0^1 f(x) x^n~dx=0$$
for all integers $n \geq 0$, then $f$ is identically zero on $[0,1]$. ({\em Hint:} This is a standard application of the Stone-Weierstrass Theorem or even just Weierstrass's theorem.)
\begin{sol}
Insert answer here.
\end{sol}
\question {\em (Kriz and Pultr \S 9.7 \# 8)}
Prove that any open set in $\R^n$ is $\sigma$-compact.
\begin{sol}
Insert answer here.
\end{sol}
\end{questions}
\end{document}