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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?

\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}
