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\title{Real Analysis}
\author{Final Exam}
\date{December 19, 2008}
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\begin{enumerate}

\item A ``totally unlucky'' number is one that contains no seven in
  any decimal expansion.  Compute the Lebesgue measure of the totally
  unlucky numbers in $[0,1]$.\footnote{From: \texttt{http://www.math.nthu.edu.tw/\~{}dhtsai/real-analysis-sample-exam-with-solution-2007-11-12.pdf}}
\hfill(10)

\item Let $(\Omega, \Sigma, \mu)$ be a measure space, $1 \leq p <
  \infty$, and $f \colon \Omega \to [0,\infty]$ a measurable
  function.  Show that
\[
  \mu \{ x \in \Omega \colon f(x) > \alpha \}
  \leq \frac1{\alpha^p} \int_\Omega f^p \, \d \mu \,.
\]
\hfill(10)

\item Let $(\Omega, \Sigma, \mu)$ be a measure space and $f \colon
  \Omega \to [0,\infty]$ a measurable function with
\[
  0 < \int_\Omega f \, \d \mu < \infty \,.
\]
Find, with justification,
\[
  \lim_{n\to\infty} 
  \int_\Omega n \, \ln
  \biggl( 1 + \Bigl(\frac{f}n \Bigr)^\alpha \biggr) \, \d \mu
\]
when
\begin{enumerate}
\item $\alpha=1$,
\item $0<\alpha<1$,
\item $\alpha > 1$.
\end{enumerate}
\hfill(5+5+5)

\item Let $1<p<\infty$.  For $f \in L^p((0,\infty))$, which for
  simplicity you may assume nonnegative, define the local average
  function
\[
  F(x) = \frac1x \int_0^x f(\xi) \, \d \xi \,.
\]
Show that $F \in L^p((0,\infty))$ with
\[
  \lVert F \rVert_p \leq \frac{p}{p-1} \, \lVert f \rVert_p \,.
\]

\emph{Hint:} Find a differential equation for $F$, then multiply by
$F^{p-1}$ and integrate.  You may assume first that $f \in
C_\text{c}^\infty ((0,\infty))$, then extend by
density.\footnote{From: \texttt{http://www.math.umn.edu/\~{}lewicka/8601-2/ex19.pdf}}
\hfill(10)

\item Let $H$ be a Hilbert space and $V \subset H$ a closed subspace.
  In linear algebra, the quotient space $H/V$ is defined as the space
  of cosets $\{ x+V \colon v \in V \}$.  (In other words, it is the
  set of equivalence classes with respect to the equivalence relation
  $x \sim y$ if $x-y \in V$.)

  How can you define an inner product on $H/V$ so that the
  \emph{quotient map} $p \colon H \to H/V$ defined by $p(x)=x+V$ is
  continuous? 
\hfill(10)

\item Let $H$ be a Hilbert space,  $\{x_n\}_{n\in\N} \subset H$ and
$x\in H$ such that 
\[
  x_n \rightharpoonup x
\]
weakly.  
\begin{enumerate}
\item Show that
\[
  \lVert x \rVert \leq \liminf_{n \to \infty} \lVert x_n \rVert \,.
\]
\item Give an example where (a) holds with strict inequality.
\end{enumerate}
\hfill(10+5)

\item Show that for $f \in L^1(\mathbb{R}^n)$,
\[
  \lim_{y\to 0}
  \int_{\mathbb{R}^n} |f(x) - f(x-y)| \, dx =0 \,.
\]
\hfill(10)

\item Compute the Fourier transform of the function $f \colon \R \to
  \R$ defined by
\[
  f(x) = x \, \e^{-x^2} \,.
\]

\emph{Hint:} Recall from class that 
\[
  \hat f(\xi) = \frac1{\sqrt{2 \pi}} \int_\R \e^{-\i \xi x} \, f(x) \,
  \d x \,;
\]
when $g(x) = \e^{-x^2}$, then
\[
  \hat g(\xi) = \frac1{\sqrt 2} \, \e^{-\xi^2/4} \,.
\]
\hfill(10)

\item Let $T \in \mathcal D'(\R)$ such that $xT=0$.  Prove that $T$
  must be a multiple of the $\delta$-distribution.

  \emph{Hint:} Choose two different test functions $\phi,\psi \in
  \mathcal D(\R)$ and apply $T$ to $\theta =
  \phi/\phi(0)-\psi/\psi(0)$.
\hfill(10)


\end{enumerate}









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