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

\doclabel{Math F651: Homework 4}
%\docauthor{Your name here.}
\docdate{Due: February 21, 2018}
\begin{document}
\begin{aproblems}



\hproblem R \& Y 3.21

\hproblem R \& Y 3.22

\hproblem R \& Y 3.23

\hproblem R \& Y 4.3

\hproblem Consider the map $I:C[0,1]\to C[0,1]$ given by
\[
(I(f))(x) = \int_0^x f(s)\; ds.
\]
Find a sequence of functions $f_n$ such that $||f_n||_{\infty}=1$
and such that $||If_n||_{\infty}\to 0$.  

Then tell me what this has to do with Example 4.10.

\hproblem For each pair $p$ and $q$ with $1\le p,q\le \infty$
determine whether the map $z\mapsto z$ from $(Z,\ell^p)$
to $(Z,\ell^q)$ is continuous.

\hproblem Consider $Z$ with the $\ell^2$ norm.  Fix $w\in \ell^p$
for some $1\le p\le \infty$ and define
\[
T(z) = \sum_{k=1}^\infty w_k z_k.
\]
Determine the values of $p$ such that $T$ is necessarily continuous.

\hproblem [Extra Credit] For each pair $p$ and $q$ with $1\le p,q\le \infty$
determine whether the map $f\mapsto f$ from $(C[0,1],L^p)$
to $(C[0,1],L^q)$ is continuous.

\end{aproblems}
\end{document}

