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\def\course{MAT 324}
\def\semester{Fall 2006}
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\begin{document}
\begin{center}
{\bf PROBLEM SET 5}
\end{center}
\begin{enumerate}
\item Does $\{ \sin(nx) \}$ converge in the $L^1$ norm on $[0, 2 \pi]$?
\item Give an example of a sequence of functions $\{ f_n\}$ which
converges to the constant zero function in $L^1$, but
so that $f_n(x)$ does not converge to zero at any
point of $[0,1]$.
\item If $f_n \to f$ in the $L^1$ norm, show that there is
a subsequence $f_{n_k}$ which converges a.e. to $f$.
\item Prove that the set of continuous functions of compact
support is dense in $L^1$.
\item For each $ 1 \leq p \leq \infty$ give an example of a
function which is in $L^p$ but not in $L^q$ for any
$q \ne p$.
\item If $f \in L^1$ show that $X_f =\{ g\in L^1: |g|\leq |f|\}$ is
a compact set in the $L^1$ topology (i.e., a sequence in this
set has a subsequence converging to a point of the set).
%\item If $f\in L^2$ let $T_nf $ be the $n$th partial sums of the
% Fourier series. Then $T_n f \to f$ in the $L^2$ norm.
% It is also true that $T_n f \to f$ a.e., but this is very
% hard to prove (one of the most famous theorems proven in
% the 20th century).
\end{enumerate}
\end{document}