\documentclass[minion]{homework}
\newcommand{\Reals}{\mathbb{R}}

\doclabel{Math F651: Homework }
%\docauthor{Your name here.}
\docdate{Due: February 7, 2018}
\begin{document}
\begin{aproblems}

\hproblem Prove that $\ell^\infty$ is complete.

\hproblem Consider the norms $||\cdot||_1$, $||\cdot||_2$, and
$||\cdot||_\infty$, on $\Reals^n$.  Since all norms on $\Reals^2$
are equivalent, there are constants $m$ and $M$ such that
\[
m||x||_1 \le ||x||_2 \le M||x||_1
\]
for all $x\in\Reals^2$.  Find, with proof, the best such constants
and make a diagram that illustrates this fact.  

Now repeat this exercise for the remaining two pairs of norms 
(i.e. the pair $||\cdot||_1$ and $||\cdot||_\infty$ and the pair
$||\cdot||_2$ and $||\cdot||_\infty$ ).

\hproblem Show that if $X$ is a Banach space and $S\subseteq X$ is a
closed subspace, then $S$ is complete (and hence a Banach space).

\hproblem R \& Y 2.10

\hproblem R \& Y 2.11(b)

\hproblem R \& Y 2.12

\hproblem R \& Y 2.13

\hproblem R \& Y 2.14

\end{aproblems}
\end{document}

