\documentclass[minion]{homework}
\usepackage{cmacros}
\def\calB{\mathcal{B}}
\begin{document}
\doclabel{Math 617: Homework 4}
\docdate{Due: February 15, 2012}
\begin{aproblems}

\hproblem\solver{David Maxwell}  Let $\tilde X$ be a completion of the normed space $X$,
and let $\phi:X\ra\tilde X$ be an isometry such that $\phi(X)$ is dense
in $\tilde X$.  
Given $\tilde x, \tilde y\in \tilde X$, let $(x_n)$ and $(y_n)$ be sequences in $X$ such that
$\phi(x_n)\ra \tilde x$ and 
$\phi(y_n)\ra \tilde y$.  We define
$$
\tilde x + \tilde y = \lim_{n\ra\infty} \phi(x_n+y_n)
$$
and
$$
\lambda \tilde x = \lim_{n\ra\infty} \phi(\lambda x_n).
$$
\begin{subproblems}
\item  Show that these limits exist and are independent of the choice
of approximating sequences. 
\item Convince yourself that it is then easy and tedious to verify
$\tilde X$ with these operations is indeed a vector space (if you 
decide prove this, don't hand it in!).
\item Show that the distance function on the metric space $\tilde X$ is
indeed a norm.
\item Show that $\phi$ is a continuous linear map.
\end{subproblems}

% \hproblem\solver{TJ Barry} 
% Suppose $X$ is a normed space that satisfies the parallelogram law.
% Show that it is an inner product space.

\hproblem\solver{Lyman Gilispie} 
Show that the completion of an inner product space $X$ is a Hilbert space.  That is, if $\phi:X\ra\tilde X$ is an isometry with dense image into
the complete space $\tilde X$, then $\tilde X$ with the normed space 
structure described in problem 1 is in fact an inner product space, and that
if $x,y\in X$,
$$
\ip<x,y>_X = \ip<\phi(x),\phi(y)>_{\tilde X}.
$$
Don't do a lot of work.

\hproblem
\begin{subproblems}
  \item Let $\phi(x)=\chi_{[0,1)}$.  Prove or disprove:
$\phi\in H^1((-1,1))$.
\item For which values of $\alpha\in(0,1]$ is 
$$
|x|^\alpha \in H^1((-1,1))?
$$
\end{subproblems}

% \hproblem\solver{Vikenty Mikheev} Suppose $P:X\ra X$ where $X$ is an inner-product space,
% and suppose $P^2=P$.  Show that $||P||=1$ if $P$ is non-trivial,
% and that the image and kernel of $P$ are orthogonal.

\end{aproblems}
\end{document}