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\begin{document}
\begin{center}
{\bf \large Math 701 Problem Set 8} \vs \\
{\bf due Friday 11/15/2013} \vs \vs
\end{center}  

\noindent
{\bf Problem 1.} Suppose $\{ P_n \}$ is a sequence of real polynomials such that $P_n \to f$ uniformly on $\RR$. What can you say about the limit function $f$? \vs

\noindent
{\bf Problem 2.} Let $\{ f_n \}$ and $g$ be Riemann integrable functions on $[0,1]$ such that $|f_n(x)| \leq g(x)$ for all $x$ and $n$. Define 
$$
F_n(x)=\int_0^x f_n(t) \, dt.
$$ 
Show that $\{ F_n \}$ has a subsequence which converges uniformly on $[0,1]$. \vs 

\noindent
{\bf Problem 3.} Let $\{ f_n \}$ be a sequence of twice differentiable functions on $[a,b]$ such that $f_n(a)=f'_n(a)=0$ for all $n$. Assume further that there is an $M>0$ such that $|f''_n(x)|\leq M$ for all $x$ and $n$. Show that $\{ f_n \}$ has a subsequence which converges uniformly on $[a,b]$. (Hint: Mean value theorem and Arzel\`a-Ascoli are helpful here.) \vs

\noindent
{\bf Problem 4.} 
\begin{enumerate}
\item[(i)]
Suppose $f:[0,1] \to \RR$ is continuous and $\int_0^1 x^n f(x) \, dx=0$ for all integers $n \geq 0$. Show that $f=0$ on $[0,1]$. \vs

\item[(ii)]  
Repeat part (i) assuming that $\int_0^1 x^n f(x) \, dx=0$ for all large enough $n$. \vs
\end{enumerate}
(Hint: In (i), the integral of $f$ against any polynomial is zero. Use the Weierstrass approximation theorem to show that $\int_0^1 f^2 =0$. For (ii), well..., isn't it obvious?) \vs

\noindent
{\bf Problem 5.} Suppose $X$ is a compact metric space. A subset $I \subset \mathscr{C}(X)$ is called an {\bit ideal} if $f,g \in I$ and $h \in \mathscr{C}(X)$ imply $f+g \in I$ and $fh \in I$. An ideal $I \neq \mathscr{C}(X)$ is called {\bit maximal} if for any ideal $J$,  the inclusions $I \subset J \subset \mathscr{C}(X)$ imply $J=I$ or $J=\mathscr{C}(X)$ (that is, there is no ideal properly between $I$ and $\mathscr{C}(X)$).  \vs
\begin{enumerate}
\item[(i)]
Verify that if $p \in X$, then  
$$
I_p = \{ f \in \mathscr{C}(X): f(p)=0 \}
$$
is a maximal ideal. \vs
\item[(ii)]
Show that every maximal ideal in $\mathscr{C}(X)$ is of the form $I_p$ for some $p \in X$. \vs
\end{enumerate}
(Hint for (ii): Let $I$ be a maximal ideal. If for every $p \in X$ there is an $f_p \in I$ with $f_p(p) \neq 0$, construct a function in $I$ which does not vanish anywhere, and conclude that $I=\mathscr{C}(X)$, which is a contradiction.)\vs 


\noindent
{\bf Problem 6.} A special case of the Stone-Weierstrass theorem shows that polynomials $\sum a_{ij} x^i y^j$ in two variables are dense in the space of all continuous functions $[a,b] \times [c,d] \to \RR$. Use the Stone-Weierstrass theorem to prove the following analog in a general product space: Suppose $X,Y$ are compact metric spaces and $f:X \times Y \to \RR$ is continuous. Then, for every $\ve>0$ there are continuous functions $f_i:X \to \RR$ and $g_i:Y \to \RR$ $(1 \leq i \leq n)$ such that 
$$
\sup_{(x,y) \in X \times Y} \left| f(x,y) - \sum_{i=1}^n f_i(x)g_i(y) \right| < \ve. 
$$ 
Thus, continuous functions of two variables can be uniformly approximated on compact sets by finite combinations of products of functions of a single variable. \vs   



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