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\begin{document}

\title{Real Analysis}
\author{Homework 6}
\date{due October 15, 2008}
\maketitle

\begin{enumerate}

\item (From Folland.) Suppose $0 < p_0 < p_1 \leq \infty$.  Find
  examples of functions $f$ on $(0,\infty)$ endowed with the Lebesgue
  measure such that $f \in L^p$ if and only if
  \begin{enumerate}
  \item $p_0 < p < p_1$
  \item $p_0 \leq p \leq p_1$
  \item $p=p_0$
  \end{enumerate}
\emph{Hint:} Consider functions of the form $x^{-a} \, \lvert \ln x
\rvert^b$. 

\item (From Folland.) If $1 \leq p < r \leq \infty$, show that $L^p
  \cap L^r$ is a Banach space with norm $\lVert f \rVert = \lVert f
  \rVert_p + \lVert f \rVert_r$, and if $p < q < r$, the inclusion map
  $\operatorname{id} \colon L^p \cap L^r \to L^q$ is continuous.

\item Write up the proof of the uniform boundedness principle (cf.\
  Lieb and Loss, Theorem 2.12) for the case $p=\infty$.

\item Let $f_k(x) = k^{\tfrac1p} \, g(kx)$ for some fixed $g \in
  L^p(\R)$ with $1<p<\infty$.  Show that $f_k$ converges weakly, but
  not strongly to zero in $L^p(\R)$.
\end{enumerate}

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