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

\def\C{{\mathbb C}}
\def\e{{\mathrm e}}
\def\d{{\mathrm d}}
\def\interior{\operatorname{int}}
\def\dist{\operatorname{dist}}
\def\Ker{\operatorname{Ker}}
\def\Range{\operatorname{Range}}
\def\N{{\mathbb N}}
\def\R{{\mathbb R}}
\def\D{{\mathcal D}}

\begin{document}

\title{Functional Analysis}
\author{Homework 5}
\date{due March 20, 2009}
\maketitle

\begin{enumerate}

\item Give an example of a Banach space $E$ having two closed
subspaces $G$ and $L$ such that $G+L$ is not closed.

\item Give an example of a Banach space $E$ with subspace $N$ such
that $\overline N \neq (N^\bot)^\bot$.

(See hints in lecture notes, p.\ 21 top paragraph.)

\item Let $E$ be a Banach space and $G$ a closed subspace.  Prove that
\[
  \dist(f, G^\bot) 
  = \sup_{\substack{x \in G\\ \lVert x \rVert \leq 1}} f(x)
\]
for any $f \in E^*$.

\item Let $A \colon \D(A) \subset E \to F$ be an unbounded linear
operator.  Recall that the \emph{graph} of $A$ is defined
\begin{gather*}
  \Gamma(A) = \{(u,Au) \colon u \in \D(A) \} \,,
\intertext{the \emph{kernel} or \emph{nullspace} of $A$ is}
  \Ker (A) = \{ u \in \D(A) \colon Au=0 \} \,,
\intertext{and the \emph{image} or \emph{range} of $A$ is}
  \Range(A) = \{ Au \colon u \in \D(A) \} \,.
\end{gather*}
Now let $X=E \times F$ with subspaces $G = \Gamma(A)$ and $L=E \times
\{0\}$.  Verify the following.
\begin{enumerate}
\item $\Ker (A) \times \{0\} = G \cap L \,,$
\item $E \times \Range(A) = G + L \,,$
\item $\{0\} \times \Ker(A^*) = G^\bot \cap L^\bot \,,$
\item $\Range(A^*) \times F^* = G^\bot + L^\bot \,.$
\end{enumerate}


\end{enumerate}

\end{document}
