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\begin{document}

\begin{center}
\begin{large}
Math 70100: Functions of a Real Variable I \\
Homework 5, due Wednesday, October 8.
\end{large}
\end{center}

{\bf Name:} Insert your name here.

\begin{questions}
\question {\em (Folland 4.7.68)}
For a space $X$, let $C(X)$ denote the continuous
functions from $X$ to $\R$ equipped with the uniform norm.
Let X and Y be compact Hausdorff spaces. 
Show that the algebra generated by functions of the form 
$f(x, y) = g(x)h(y)$, where $g \in C(X)$ and $h \in C(Y)$
is dense inside of $C(X \times Y)$. 
\begin{sol}
Insert answer here.
\end{sol}

\question {\em (Folland 4.7.69)}
Let $A$ be a nonempty set, and let $X = [0, 1]^A$. Show that the algebra generated by the
coordinate maps $\pi_a : X \to [0, 1]$ and the constant function 
${\mathbf 1}$ is dense in $C(X)$.
\begin{sol}
Insert answer here.
\end{sol}

\question {\em (Rephrased Lang III \S 4 \# 19)}
Let $\R_{\geq 0}=\{x \in \R~:~x \geq 0\}$, and let $C_0(\R_{\geq 0})$ denote the
continuous real-valued functions on $\R_{\geq 0}$ which vanish at infinity.
Prove that $C_0(\R_{\geq 0})$ is the uniform closure of the collection of all functions
of the form $e^{-x} p(x)$, where $p$ is a polynomial.
({\em Lang's Hint; note he phrases this question differently:} First show that you can
approximate $e^{-2x}$ by $e^{-x} q(x)$ for some polynomial q(x), by using Taylor's formula with remainder. 
If $p$ is a polynomial, approximate $e^{-nx} p(x)$ by $e^{-x} q(x)$
for some polynomial $q$.)
\begin{sol}
Insert answer here.
\end{sol}

\end{questions}

\end{document}
