\documentclass[minion]{homework}
\usepackage{cmacros}
\def\calB{\mathcal{B}}
\DeclareMathOperator{\Ker}{{\rm Ker}}
\DeclareMathOperator{\Image}{{\rm Image}}
\begin{document}
\doclabel{Math 617: Homework 9}
\docdate{Due: March 28, 2012}
\begin{aproblems}
\aproblem  Let $A$ be a continuous operator on the Hilbert space $X$.
Show $(A^*)^*=A$.

\aproblem D\&M 4.4b, c  (Note that we effectively already proved part a early on in the semester.)

\aproblem D \& M 4.23

\aproblem  Suppose $A$ is a continuous operator on the Hilbert space $X$.
Show
$$
\Image(A)^\perp = \Ker(A^*)
$$
and
$$
\Ker(A)^\perp = \overline{\Image(A^*)}
$$

\aproblem Let $I$ be a bounded open interval and let $f\in L^1(I)$.
Suppose $\int f\psi=0$ for all $\psi\in\mathcal{D}(I)$.
\begin{subproblems}  
\item Conclude that $\int_J f = 0$ for all closed intervals $J\subseteq I$.
(Hint: Homework 8, problem 6).
\item Conclude that $\int_E f=0$ for all measurable sets $E\subseteq I$.
(Hint: Approximate $E$ with finitely many closed intervals and take advantage
of Carothers 18.17, which you have already proved.)
\item Conclude that $f=0$ almost everywhere on $I$.
\end{subproblems}
\end{aproblems}
\end{document}