\documentclass[12pt]{article}
\usepackage{amsmath,amsfonts,fullpage}

\def\C{{\mathbb C}}
\def\e{{\mathrm e}}
\def\d{{\mathrm d}}
\def\N{{\mathbb N}}
\def\R{{\mathbb R}}
\def\Q{{\mathbb Q}}

\begin{document}

\title{Real Analysis}
\author{Homework 9}
\date{due November 12, 2008}
\maketitle

\begin{enumerate}

\item Assume that $(\Omega, \Sigma, \mu)$ is a measure space, where
  $\Sigma$ is generated by an algebra $\mathcal A$.  Assume that the
  measure space is $\sigma$-finite in the strong sense that
\[
  \Omega = \bigcup_{j=1}^\infty A_j
\]
with $A_j \subset \mathcal A$ and $\mu(A_j)<\infty$.  Let $f \in
L^p(\Omega)$ for $1 \leq p < \infty$.

Show that for every $\varepsilon >0$ there exists a really simple
function $g$ such that
\[
  \lVert f-g \rVert_p < \varepsilon \,.
\]

\emph{Note:} The case $p=1$ was proved in class.  To reduce the
general case to the case $p=1$, you may refer to Question~1 from
Homework~8. 

\item Show that if $f \in L^p(\R^n)$ for $1 \leq p < \infty$ and $g
  \in C_{\mathrm{c}}^\infty(\R^n)$, then $f*g \in C^\infty(\R^n)$ and
  for every multi-index $\alpha$, 
\[
  D^\alpha (f*g) = (D^\alpha g) * f \,.
\]

\item Let $H$ be a Hilbert space and $K \subset H$ be closed and
  convex.  Then for every $x \in H \setminus K$ there exists $y \in K$
  such that 
\[
  \lVert x-y \rVert = \operatorname{dist} (x,K) \,.
\]
Moreover, for every $z \in K$,
\[
  \operatorname{Re} \langle z-y,x-y \rangle \leq 0 \,.
\]

\emph{Note:} This is the Hilbert space version of Lieb and Loss, Lemma
2.8 (Projection onto convex sets).

\item Show that every Hilbert space is dual to itself with the obvious
  identification that $x \in H$ gives rise to $\ell \in H^*$ via
  $\ell(y) = \langle x,y \rangle$ for every $y \in H$.

\emph{Note:} This is the Hilbert space version of Lieb and Loss,
Theorem 2.14 (Dual of $L^p$).  The proof is analogous and needs the
result from Question~3 above.


\end{enumerate}

\end{document}
