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\topmatter
\title
\bf \bf Ph.D. Comprehensive Examination\\
Real Analysis Section %Real Analysis Qualifying Exam
\endtitle
\author
\bf January 1996
\endauthor
\endtopmatter
\document




\t{Part I}{Do three (3) of these problems.}

\s{I.1} {Let $\{a_n\}$ be a sequence of real numbers with the following
property: there is a constant $0 < K < 1$ such that 
 $$
 |a_{n+2} - a_{n+1}| \leq K|a_{n+1} - a_n| \quad \text{for all }n \geq N_0.
 $$
 Prove that $\{a_n\}$ converges.}
 
\s{I.2} {Let $f: [a,b] \to \Bbb R$ be a continuous function and $x_1, \dots,
x_n \in [a,b]$. Show that there exists $z \in [a,b]$ such that
 $$
 f(z) = \frac{f(x_1) + \cdots + f(x_n)}{n}.
 $$}

\s{I.3}{Give an example of a function $f \in L^p(\Bbb R)$, $p \geq 1$, such
that
 $$
 \lim_{x\to \infty} f(x) \ne 0.
 $$}

\s{I.4} {\roster
 \runinitem Let $\{f_n\}$ be a subsequence of $L^1(\Bbb R)$ such that $\sum_{n=1}^\infty
\|f_n\|_{1} < \infty$. Show that $\sum_{n=1}^\infty f_n$ converges absolutely
a.e.
 \item Let $(X,\Cal A, \mu)$ be a measure space and let $\{A_n\}$ be a
subsequence of $\Cal A$. Show that if $\sum_{n=1}^\infty \mu(A_m) < \infty$
then $\mu(\limsup A_n) = 0$, where
$\limsup A_n = \bigcap_{m=1}^\infty\bigcup_{n=m}^\infty A_n$.
 \endroster}

\vglue\baselineskip
\t{Part II}{Do two (2) of these problems.}

\s{II.1}{ Let 
 $$
  F(y) = \int_0^\infty e^{-2x} \cos (2xy) dx, \quad y \in \Bbb R.
 $$
 Show that $F$ satisfies the differential equation
  $$
  F'(y) + 2y F(y) = 0.
  $$
  Justify the differentiation under the integral sign.}

\s{II.2}
{Let $f$ be a real valued function defined on a closed bounded interval
$[a,b]$. Establish the following:
 \roster
 \item If $f$ is continuous, $f$ need not be of bounded variation. Consider
  $$
  f(x) = \left\{
  \aligned 
  x\sin{\frac 1 x} & \quad  \text{if } 0 < x \leq 1 \\
  0\quad &\quad \text{if } x = 0
  \endaligned
  \right.
  $$
 \item If $f$ satisfies a Lipschitz condition, that is, $|f(x) - f(y)| \leq
M|x-y|$ for some positive number $M$ and all $x$, $y \in [a,b]$, then $f$ is
absolutely continuous.
 \item If $f'$ exists everywhere and is bounded on $[a,b]$, then $f$ is
absolutely continuous.
 \endroster}

\s{II.3}
{Let $H$ be a Hilbert space and $y_0 \in H$. Show that there exists $\Lambda
\in H^*$ (bounded linear functional on $H$) different from zero such that
 $$
 \Lambda(y_0) = \|\Lambda\|_{H^*}\|y_0\|.
 $$
 (Hint: Either apply the Hahn-Banach theorem with the sublinear functional
$p(x) = \|x\|$, or construct a bounded linear functional in terms of $y_0$).
}

\enddocument

euclid%