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\begin{center}
{\bf Spring 2002 Ph.D Qualifying Exam in Real Analysis}\end{center}
\noindent{\em Time 3 hours, closed book, no notes. Answer three
questions from each of the parts A \& B.} \vskip 20 pt
\noindent Part A
\be
\item Prove Egoroff's Theorem namely: Suppose $(X,\cal{M},\mu)$
is a measure space and $\mu(X)<\infty$. Let $\{f_n\}$ be a sequence
of measurable functions such that $f_n\rightarrow f$ (a.e.).
Show that for any $\epsilon>0$, there is a measurable set $E$
with
$\mu(E)<\epsilon$ such that $f_n\rightarrow f$ uniformly on
$X\backslash E$.
\item Let $X$ be a complete metric space. Suppose $\cal{F}$ is a
family of continuous functions on $X$ with the property that for
each $x\in X$, there is a constant $M_x$ such that $|f(x)|\leq
M_x$ for all $f\in \cal{F}$. Show that there exists a non-empty
open set $O\subset X$ and a constant $M$ such that $|f(x)|\leq M$
for all $f\in\cal{F}$ and $x\in O$.
\item Let $f_n$ be a sequence of continuous real-valued
functions on the closed interval $[a,b]$ with $f_n(x)\leq
f_{n+1}(x)$, and with $\lim_{n\rightarrow\infty}f_n(x)=f(x)$ for
each $x\in[a,b]$. Prove that if $f$ is continuous on $[a,b]$, then
the convergence is uniform.
\item Suppose $\sum_{k=0}^{\infty}a_k$ converges. Let
$f(x)=\sum_{k=0}^{\infty}a_kx^k$ and
$g(x)=\sum_{k=0}^{\infty}ka_kx^{k-1}$.
\be\item Show that both series converge uniformly on
$[-\rho,\rho]$ for any $\rho, \newline 0<\rho<1$.
\item Show that $f'(x)=g(x)$.
\item Show that $\lim_{x\rightarrow
1^-}f(x)=\sum_{k=0}^{\infty}a_k$.\ee
\item Let $X$ be a metric space with metric $\rho$. Show that $X$ is
totally bounded if and only if every sequence $\{x_n\}$ in $X$ has
a Cauchy subsequence.
\item Let $\phi_n(x)=\sqrt{n}e^{-n^2|x|}$ and let
$f(x)=\sum_{n=0}^{\infty}\phi_n(x-r_n)$ where $\{r_n\}$ is an
enumeration of rational numbers in $\mathbb{R}$. Show: \be\item
$f(x)\in L^1(\mathbb{R},m)$ and compute $\int_{\mathbb{R}}f(x)dx$
\item $f(x)$ is unbounded in any open interval $(a,b)$.\ee\ee
\newpage
\noindent Part B
\be
\item Show that
$$\lim_{n\to \infty}\int_{0}^{n} \left(1 - \frac{x}{n}\right)^{n}dx =
\lim_{n\to \infty}\int_{0}^{n} \left(1+\frac{x}{n}\right)^{n}
e^{-2x}dx=1.$$
\item \be \item State the Stone-Weierstrass theorem.
\item If $f$ is a continuous function on $[0,1]$ such that
$\int_{0}^{1}x^{n}f(x)dx=0$ for $n=1, 3, 5, \cdots$, then show
that $f(x)=0$ for all $x\in [0,1]$.
\item Show that the algebra generated by the set $\{1, x^{2}\}$
is dense in $\mathcal{C}([0,1])$ but fails to be dense in
$\mathcal{C}([-1,1])$. \ee
\item Suppose that $f$, a function defined on an open interval
$(a,b)$,
satisfies the intermediate value theorem i.e., if $f$ assumes the
values $y_1,y_2$, it assumes all values between $y_1,y_2$. Show
that if $f$ is not continuous, it assumes some value infinitely
often.
\item Let $f(x)$ be a continuous real-valued function on a compact
metric space $(X,\rho)$. Show that $f$ is uniformly continuous.
\item Let $K(x,y)$ be continuous on the closed unit square
$[0,1]{\rm x}[0,1]$ in $\mathbb{R}^2$. For $f\in
\mathcal{C}([0,1])$, we define
$$g(x)=T(f)(x) = \int_0^1K(x,y)f(y)dy.$$
Prove that if $\{f_n\}$ is a sequence in $\mathcal{C}([0,1])$ with
$\|f_n\|\leq M$ for all $n$, there is a subsequence $\{n_k\}$ such
that the sequence $\{g_{n_k}\}$ converges uniformly.
\item Let $(X,\rho)$ be a metric space and $A\subset X$ be a
non-empty subset. Let $\rho(x,A)=\inf\{\rho(x,z)\mid z\in A\}$,
the distance from $x$ to $A$. \be\item Show that $\rho(x,A)$ is
uniformly continuous on $X$. In fact\newline
$|\rho(x,A)-\rho(y,A)|\leq\rho(x,y).$
\item Now show that $\rho(x,A)=0$ if and only if $x\in\overline{A}.$\ee
\ee
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