\documentclass[minion]{homework}
\usepackage{cmacros}
\def\calB{\mathcal{B}}
\begin{document}
\doclabel{Math 617: Homework 6}
\docdate{Due: February 29, 2012}
\begin{aproblems}

Many problems on the current assignment use the Baire category
theorem, which is not in your text.  We presented two versions
in class today.  Here's a third formulation that might be helpful.

\proposition Let $X$ be a complete metric space.
  If $\{F_i\}$ is a sequence of closed sets in $X$ and
  if $X=\cup_{i} F_i$ then at least one $F_i$ has non-empty interior.
\NoProof
This is little more than a recapitulation of the statement that
a complete metric space is not meager.

\hproblem\solver{David Maxwell} Suppose that $X$ is a real normed vector space
that satisfies the parallelogram law.  We will show that
\[
\ip<x,y>=\frac{1}{4}\left[||x+y||^2-||x-y||^2\right]
\]
is an inner product on $X$ that generates its norm.

Let $a,b,y\in X$.
\begin{subproblems}
  \item Use two applications of the parallelogram law to show
  $$
  \ip<a+b,y>= 2\ip<a,y/2>+2\ip<b,y/2>.
  $$
  \item Conclude that $\ip<a+b,y>=\ip<a,y>+\ip<b,y>$.
  \item Show that $\ip<k a,y>=k\ip<a,y>$ for all $k\in\Nats$, and then 
  for all $k\in\Rats$.
  \item Justify the conclusion 
  that $\ip<\alpha a,y>=\alpha \ip<a,y>$ for all $\alpha\in\Reals$.
\end{subproblems}

\hproblem\solver{TJ Barry} Show that a complete metric space without any 
isolated points is uncountable.

\hproblem\solver{Vikenty Mikeev} 
Let $X$ be a Hilbert space and suppose $P:X\ra X$, $P^2=P$,
and $||P||=1$.  Show that $P$ is the orthogonal projection on to
$P(X)$.

\hproblem\solver{David Maxwell} 
Show that the plane is not a countable union of lines.

\hproblem\solver{Lyman Gilispie}
Show that a normed vector space with an  countably infinite algebraic basis is not complete.

\hproblem\solver{Will Mitchell} 
Let $X$ be a normed space and $S\subseteq X$. Show that
if for each $f\in  X^*$ the set $\{f(x):x\in X\}$ is bounded,
then $S$ is bounded.

\end{aproblems}
\end{document}