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\centerline{\bf Ph.D. Real Analysis Qualifying Examination}
\bigskip
\begin{center}
January 24, 2009\\
Examiners: Rao Nagisetty and Denis White
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\noindent {\bf Instructions:} Do six of the following questions. No materials are allowed.
\begin{enumerate}
\item %1
\begin{enumerate}
\item
State the Baire Category Theorem. If you use the terminology ``first category'' or ``second category''
then you should define those terms.
\item
Suppose that $\{f_{\lambda}: \lambda \in \Lambda\}$ is a collection of continuous complex valued functions defined on
a complete metric space $X$. Suppose further that, for every $x \in X$, there is $\epsilon_{x}>0$ so
that $|f_{\lambda}(x)|>\epsilon_{x}$ for every $\lambda \in \Lambda$. Show that there is $\epsilon>0$ and an open set
$U \subseteq X$ so that $|f_{\lambda}(x)|>\epsilon$ for all $x\in U$ and $\lambda \in \Lambda$.
\end{enumerate}
\item %2
\begin{enumerate}
\item
Define equicontinuity.
\item
State the Arzela Ascoli Theorem.
\item
Suppose that $f_n$, $n \in \mathbb{N}$ is a sequence of differentiable functions
defined on $[0,\infty)$ such that
\begin{eqnarray*}
\frac{d}{dx}f_n(x) &=& (1+x^2 + f_n(x)^4)^{-1/2} \hspace{2mm} \mbox{ for } \hspace{2mm} x>0 \\
f_n(0) &=& \sin n
\end{eqnarray*}
Show that, for any $b>0$, there is a subsequence of the $f_n$ which converges uniformly on $[0,b]$.
\end{enumerate}
\item %3
Suppose that $\mu$ is a Lebesgue Stieltjes measure (which means $\mu$ is a measure defined on the
Borel subsets of $\mathbb{R}$
and is finite valued on bounded sets). Suppose that $F$ is a corresponding distribution function
which means that
$\mu((a,b])=F(b)-F(a)$ whenever $a**0$ is given show that there are finitely many constants $a_k$, $1 \le k \le n$, so that
$$
|f(x)- \sum_{1 \le k \le n} a_k e^{-kx}|< \epsilon \hspace{2mm}\mbox{ for all } \hspace{2mm} x \in I
$$
\item
Show that the statement in Part (a) is false if the interval $I$ is $I=[0,\infty)$.
\item
Suppose that $f$ is continuous on $I=[0,\infty)$ and $\lim_{x \to \infty}f(x)=0$. Is the conclusion in Part (a) true
for such $f$?
\end{enumerate}
\item
\begin{enumerate}
\item
Show that the series $\sum_n \frac{1}{1+n^3x^2} $ converges uniformly for $x$ in any compact subinterval of (0,1]
but does not converge uniformly on (0,1] itself.
\item
Define $f(x) =\sum_n \frac{1}{1+n^3x^2} $ for $0**