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\topmatter
\title
\bf \bf Ph.D. Comprehensive Examination\\
Real Analysis Section %Real Analysis Qualifying Exam
\endtitle
\author
\bf Fall 1994
\endauthor
\endtopmatter
\document




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

\s{I.1} {\roster \runinitem "(a)" Give an example of a function $f(x)$ such that
$\lim_{m\to \infty} \int_0^m f(x)\, dx$ exists, but $\lim_{m\to
\infty}\int_0^m|f(x)|\,dx$ does not exist.\endroster
\hskip1cm (b) Give an example of a function $f(x)$ such that
$\lim_{\varepsilon \to 0} \int_0^\varepsilon f(x)\, dx$ exists, but
$\lim_{\varepsilon \to 0}\int_0^m|f(x)|\,dx$ does not exist.
} 
\s{I.2} {Give an example of a countable dense subset for each of the
following:
 \roster
 \item"(a)" $\ell^2$ (in the $\ell^2$ norm).
 \item"(b)" $L^2[0,1]$ (in the $L^2$ norm).
 \item"(c)" $L^1[0,1]$ (in the $L^1$ norm).
 \endroster
}

\s{I.3}{Let $f_n$ be a sequence of absolutely convergent continuous functions
in $[a,b]$ such that $f_n(a) = 0$. Suppose that $f'_n$ is a Cauchy sequence in
$L^1[a,b]$. Show that there exists $f$, absolutely continuous in $[a,b]$, such
that $f_n \to f$ uniformly in $[a,b]$.}

\s{I.4} {Let $f$ be a non-negative function in $\Bbb R$. Suppose that the
double integral 
 $$
 \iint _{\Bbb R^2} f(4x)f(x-3y) dxdy = 2.
 $$
 Calculate $\int_{-\infty}^\infty f(x) dx$.}

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

\s{II.1}{ Prove: Every $L^1$ function is continuous in the $L^1$ norm, that
is,
 $$
 \lim_{h\to 0}\int_0^1 |f(x+h) - f(x) |dx = 0
 $$
 Note: You may assume $f$ vanishes outside $[0,1]$.}

\s{II.2}
{Given a collection of closed subintervals of $[0,1]$ such that any two of the
subintervals have a point in common, prove that all of them have a point in
common.}

\s{II.3}
{Let $p > 1$, and $\frac 1 p + \frac 1 q = 1$. Show that if $g \in L^q[0,1]$,
then 
 $$
 \ell(f) = \int_0^1 f(x)g(x)/,dx
 $$ 
 is a continuous linear functional on $L^p[0,1]$. Find $\|\ell\|$}

\enddocument

euclid%