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\begin{center} {\bf Ph.D. Qualifying Exam in Real Analysis\\
Spring 2000, Time: 3 hours, closed book, no notes.}\end{center}
\noindent Answer any six questions to get full credit, providing
as much detail as time permits.
\be
\item \be\item Let $f_n,g_n,f,g$ be Lebesgue summable,
$f_n\rightarrow f, g_n\rightarrow g$ almost everywhere,
$|f_n|\leq g_n$ and $\int g_n\rightarrow\int g.$ Show that $\int
f_n\rightarrow\int f.$\item Also show that
$\int|f_n|\rightarrow\int |f|$ if and only if
$\int|f_n-f|\rightarrow 0$.\ee
\item Let $X$ be a complete metric space and $X=\cup F_n$,
countable union of closed sets $F_n$. then at least one of $F_n$
has non-empty interior.
\item Assume that $f$ is twice continuously differentiable in the
interval $(a,\infty)$ and $M_0,M_1,M_2$ are the least upper
bounds of $f,f',f''$ respectively. show that $$M_1^2\leq M_0M_2.$$
In what way should the inequality be modified in case of a finite
interval.
\item Show that $$\int_0^{\infty}\frac{\cos
x}{1+x}dx=\int_0^{\infty}\frac{\sin x}{(1+x)^2}$$ where the left
integral converges absolutely and the right integral converges
only conditionally. Prove all assertions.
\item Assume $f$ is continuously differentiable in the closed
interval $[a,b]$ and $f(a)=0,f(b)=0,\int_a^bf(x)^2dx=1$. Show that
$$\int_a^bxf(x)f'(x)dx=-\frac12$$ and
$$\int_a^bf'(x)^2dx\int_a^bx^2f(x)^2dx>\frac14.$$
\item\be\item Prove Riemann-Lebesgue lemma: If $f(x)$ is Lebesgue
summable on an interval $[a,b]$, then
$\int_a^be^{inx}f(x)dx\rightarrow 0$ as $n\rightarrow\infty$.
\item Let $\{n_k\}$ be an increasing sequence of positive integers
and let $E$ be the set of all $x$ for which the sequence $\{\sin
n_kx\}$ converges. Show that the measure of $E$ is zero. [Hint:
You may assume that $f$ can be approximated by continuously
differentiable functions $g$ so that $\int|f-g|dx$ is small.]\ee
\item A mapping $f:\mbR\rightarrow\mbR$ is open if the image of
any open interval is open. Show that any continuous open map is
monotone.
\item Suppose that $f$ is Lebesgue integrable on
$(-\infty,\infty)$. Show that
$$\lim_{n\rightarrow\infty}f(x+n)=0$$ almost everywhere and the
function $$\int_{-\infty}^xf(t)dt$$ is continuous.
\item Show that $$\sum_{n=1}^{\infty}\frac{(-1)^{[\sqrt n]}}n$$
converges.
\item\be\item Gauss' Second Mean Value Theorem is usually stated
as follows: Assume $f,g$ are Riemann integrable on $[a,b]$ and
$g$ is monotone. Then there exists a $\xi$ in $[a,b]$ such that
$$\int_a^bf(x)g(x)dx=g(a)\int_a^{\xi}f(x)dx+g(b)\int_{\xi}^bf(x)dx.$$
Proof in this generality is quite involved. But assuming that $g$
is continuously differentiable, prove Gauss' Theorem.
\item Further if $g$ is decreasing and non-negative, show that
$\int_a^bf(x)g(x)dx=g(a)\int_a^{\xi}f(x)dx.$\ee
\item State Stone-Weierstrass Theorem and using it or otherwise,
prove that any continuous even function on $[-\pi/2,\pi/2]$ can
be approximated by linear combinations of
$\{1,\cos^2x,\cos^22x,\ldots\cos^2kx,\ldots\}$. \ee
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