\documentclass[minion]{homework}
\usepackage{cmacros}
\def\calB{\mathcal{B}}
\DeclareMathOperator{\Ker}{{\rm Ker}}
\begin{document}
\doclabel{Math 617: Homework 7}
\docdate{Due: March 7, 2012}
\begin{aproblems}

\hproblem
Let $X$ be a Hilbert 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.

\hproblem  Let $X$ be a vector space.  If $W$ is a subspace
of $X$ we can put an equivalence relation on $X$ by
$x\sim y$ if $x-y\in W$ (or alternatively if $x=y+w$ for some $w\in W$).
We write equivalence classes as $x+W$ rather than $[x]$.  
The set of equivalence classes is denoted $X/W$.  We can 
put a vector space structure on $X/W$ by $(x+W)+(y+W)=(x+y)+W$
and $\lambda (x+W)=(\lambda x)+W$.  You are invited to prove
to yourself (but not to me) that these operations are well-defined and that
$X/W$ becomes a vector space with these operations.

Now suppose further that $X$ is a normed space and $W$ is a closed subspace
of $X$.
\begin{subproblems}
  \item Show that $X/W$ is a normed space with $||x+W||_{X/W}=\inf_{y\in x+W} ||y||_X$.
  \item Show that if $X$ is a Banach space, then so is $X/W$. (Hint: Use Banach's characterization of complete spaces.)
\end{subproblems}

\hproblem Suppose $T:X\ra Y$ is a continuous surjective 
linear map between Banach spaces.  Show that $X/\Ker T$ is isomorphic
as a Banach space to $Y$.  That is, there is a continuous linear bijection
$S:X/\Ker T\ra Y$ that has a continuous inverse.

\hproblem  Suppose $X$ is a closed subspace of $L_2[0,2]$ 
and that for every $f\in L_2[0,1]$ there is an $F\in L_2[0,2]$
such that $F|_{[0,1]}=f$.  Show that there is a $c>0$ such that
we can pick $F$ such that $||F||_{L_2[0,2]}\le c ||f||_{L_2[0,1]}$.

\end{aproblems}
\end{document}