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\centerline{\bf Ph.D. Real Analysis Qualifying Examination}
\bigskip
\begin{center}
January  24, 2009\\
Examiners: Rao Nagisetty and Denis White
\end{center}
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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 $a<b$.  Suppose further that $F$ is continuously differentiable on
some interval $I$ with derivative $F'$.  Show that for any bounded Borel measurable function $g$ defined on $I$
and any $a,b \in I$,
$$
 \int_{[a,b]} g d\mu = \int_{[a,b]} gF' dm
$$
where $m$ is Lebesgue measure.
\item
Suppose that $f$ is a continuous function defined on a interval $I \subseteq \mathbb{R}$.
 \begin{enumerate}
  \item
  If $I=[0,1]$
 and $\epsilon>0$ 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 $0<x \le 1$.  Show that $f$ is not bounded.
  \item
Is $f \in L^1((0,1])$?
  \end{enumerate}
 \item
  \begin{enumerate}
    \item
   Show that a compact subset of a metric space is closed and bounded.
   \item
   Is the converse true?  Prove or give a counter example.
  \end{enumerate}
  
  \item If $f$ is real valued and satisfies the intermediate value property
on $[a b]$ and further it is not continuous, then it must assume
some value infinitely often.  (Recall that $f$ has the intermediate
value property on $[a,b]$ if, whenever $(c,d)$ is a subinterval of $[a,b]$, 
$f$ assumes every value strictly between $f(c)$ and $f(d)$ on the 
interval $(c,d)$.)
 
\item Show that any convex function defined on an open interval is 
differentiable
almost everywhere.
\item 
\begin{enumerate} \item Define the convergence of an infinite product $\ds\Pi_{n=1}^\infty(1+u_n)$.
\item Show that it converges if the series $\ds\sum_{n=1}^\infty|u_n|$ is convergent.
\end{enumerate}
\end{enumerate}
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