\documentclass[minion]{homework}
\usepackage{cmacros}
\def\calB{\mathcal{B}}
\begin{document}
\doclabel{Math 617: Homework 3}
\docdate{Due: February 8, 2012}
\begin{aproblems}
\hproblem \solver{David Maxwell}  
Suppose $X$ is a normed space
and $W\subseteq X$ is a subspace.  Show that $\overline{W}$
is a subspace.

\hproblem \solver{TJ Barry} 
Suppose $X$ is a Banach space. Let $I:X\ra X$ be the 
identity operator.
\begin{subproblems}
\item Show that if $T\in\calB(X,X)$ and $||I-T||<1$, then
$T$ is invertible.  Hint:  Consider the Taylor series for $1/(1-x)$.
\item Show that the set of invertible elements of $\calB(X,X)$ is open.
\end{subproblems}

\hproblem \solver{Lyman Gilispie}
Let $X$ and $Y$ be  vector spaces and let $\mathcal{B}=\{x_\alpha\}_{\alpha\in I}$ be a basis for $X$. 
Given a map $F:\mathcal{B}\ra Y$, show that there exists a unique
linear map $T:X\ra Y$ such that $T|_{\mathcal{B}}=F$.

Moral:  A linear map is completely specified by it is action on a basis.

\hproblem \solver{David Maxwell}
\begin{subproblems}
  \item Given an $n$-dimensional vector space $X$ over $\Reals$, show
that there exists a (real) linear isomorphism between $X$ and $\Reals^n$.
  \item If $X$ is a real or complex normed vector space and $T:\Reals^n\ra X$
  is an injective real-linear map, show that $||\cdot||_T$ on $\Reals^n$
  defined by $||x||_T = ||Tx||_X$ is a norm on $\Reals^n$.
  \item Show that if $X$ is a real or complex normed vector space, then
  any two norms on $X$ are equivalent.
\end{subproblems}
  
\hproblem \solver{Vikenty Mikheev}
Suppose $X$ is a finite dimensional normed vector space and $W$ 
is a proper subspace of $X$.  Show that there exists an $x\in X$ with
$||x||=1$ and $d(x,w)\ge 1$ for all $w\in W$.

\hproblem \solver{Will Mitchell} 
Suppose $p>1$ and $\frac{1}{p}+\frac{1}{q}=1$.
Given $y\in \ell_q$, and $x\in\ell_p$ let
$$
T_y(x) = \sum_{k=1}^\infty x_k \overline{y_k}.
$$
We will prove in class that $T_y$ is well defined, linear, and continuous.
\begin{subproblems}
  \item Show that $||T_y||=||y||_{\ell_q}$ for all $y\in\ell_q$.
  \item Given $S \in \mathcal{B}(\ell_p,\Cplx)$, show that $S=T_y$ for
  some $y\in \ell_q$.
  \item Show that $\ell_p$ is complete for all $p\in (1,\infty)$.
\end{subproblems}


  
\end{aproblems}
\end{document}