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

\def\C{{\mathbb C}}
\def\e{{\mathrm e}}
\def\i{{\mathrm i}}
\def\d{{\mathrm d}}
\def\N{{\mathbb N}}
\def\R{{\mathbb R}}
\def\Z{{\mathbb Z}}
\def\T{{\mathbb T}}

\begin{document}

\title{Real Analysis}
\author{Homework 12}
\date{due December 3, 2008}
\maketitle

\begin{enumerate}

\item Show that $\{ \e^{\i k x} \colon k \in \Z \}$ separates points
  on $\T$.

\item Let $\Omega \subset \R^n$ be open and $\mu$ a (positive) Borel
  measure such that $\mu(K) < \infty$ for all compact $K \subset
  \Omega$.  Show that
\[
  T_\mu (\phi) = \int_{\Omega} \phi \, \d \mu
\]
defines a distribution $T_\mu \in \mathcal D'(\Omega)$.

\item Let $\Omega \subset \R^n$ be open.  Let $f_j$ be a sequence in
  $W_{\operatorname{loc}}^{1,1}(\Omega)$, the space of
  $L^1_{\operatorname{loc}}(\Omega)$ functions whose first order
  distributional derivatives are also
  $L^1_{\operatorname{loc}}(\Omega)$, such that
\[
  f_j \to f \qquad \text{in } L^1_{\operatorname{loc}} (\Omega)
\]
and, for fixed $i \in 1, \dots, n$, 
\[
  \partial_i f_j \to g \qquad \text{in } L^1_{\operatorname{loc}} (\Omega)
\]
where $\partial_i f$ denotes the distributional partial derivative in
the $i$th coordinate direction.  Show that
\[
  \partial_i f = g \,.
\]

\emph{Hint:} See comments in Lieb and Loss, Section 6.7.


\item Let $j \in L^1(\R^n)$ with 
\[
  \int_{\R^n} j \, \d x = 1 \,,
\]
and set
\[
  j_\varepsilon(x) = 
  \frac1{\varepsilon^n} \, j \Bigl( \frac{x}\varepsilon \Bigr) \,.
\]
Show that
\[
  j_\varepsilon \to \delta \qquad \text{in } \mathcal D'(\R^n)
\]
as $\varepsilon \to 0$ (where $j_\epsilon$ is identified with a
distribution in the canonical way).

\end{enumerate}

\end{document}
