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\begin{document}

\centerline {\bf Math 523b - Homework 3} 

\bigskip

\begin{enumerate}


\item Problem 2, p. 154 in Folland. 

\item Problem 9, p. 155 in Folland. 

\item Problem 21, p. 160 in Folland. 

\item Problem 22, p. 160 in Folland. 

\item Problem 25, p. 160 in Folland. 

\item Problem 34, p. 164 in Folland. 

\item Problem 37, p. 165 in Folland. 

\item Problem 39, p. 165 in Folland. 

\item This problem is a special case of a general theorem we will
prove later. Let $c_0$ denote the Banach space of sequences 
which converge to $0$ with the norm
$$
|| (x_n)_{n=1}^\infty||_\infty = \sup_n |x_n|
$$
Let $l^1$ be the Banach space of absolutely summable sequences
with the norm
$$
|| (x_n)_{n=1}^\infty||_1 = \sum_{n=1}^\infty |x_n|
$$
Our goal is to show that the dual of $c_0$ can be identified with 
$l^1$ in a natural way. 

\no (a) Given $x=(x_n)_{n=1}^\infty \in l^1$, define a linear functional
$\phi_x$ on $c_0$ by 
$$
\phi_x((y_n)_{n=1}^\infty) = \sum_{n=1}^\infty x_n y_n
$$
Prove that $\phi_x$ is a bounded linear functional whose norm equals
$||x||_1$. Thus the map $x \ra \phi_x$ is an isometry of 
$l^1$ into the dual of $c_0$. \

\no (b) Prove that this isometry is onto. 


\end{enumerate}
\end{document}
\end

