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\noindent Homework 3

\noindent Math 336, Winter `00

\noindent Due Friday, February 11
  
\vskip .2in

\noindent From the textbook:
\begin{itemize}
\item Page 202: 1, 6  
\item Page 209: 4abc (give a counterexample if the space is not complete),
                     
\end{itemize}

\vskip .2in
\noindent{\bf Other Problems:}
\vskip .2in

\noindent A.  Consider the set $\mathcal{M}$ of continuous 
functions $f:[0,1]\to\mathbf{R}$ with the metric 
$$
\rho(f,g) = ||f-g||_1 = \int_0^1 |f(x) - g(x)|\,dx
$$
is not complete.  Give an example of a sequence 
$\{f_n\}_{n=0}^\infty\subset\mathcal{M}$ that 
converges in this metric but not in the
metric $||f-g||_\infty$.
(Hint: find a sequence that converges in this metric to a 
discontinuous function)
\vskip .2in

\noindent B.  Two metrics $\rho_1,\rho_2$ on a set $\mathcal{M}$ are
called {\it comparable} or (in the book's terminology) {\it uniformly
equivalent} if there exist constants $c_1,c_2 > 0$ such that
$$
c_1\rho_1(x,y) < \rho_2(x,y) < c_2\rho_1(x,y)
$$
for every $x,y\in\mathcal{M}$.  
\begin{description}
\item[(i)] Show that if $\rho_1,\rho_2$ are comparable metrics on 
$\mathcal{M}$, then a sequence $\{a_n\}_{n=1}^\infty$
converges with respect to the metric $\rho_1$ if and only if it
converges with respect to the metric $\rho_2$.  
\item[(ii)] Let $\rho_1, \rho_{max}, \rho_2$ be the metrics 
described in the textbook for $\mathbf{R}^n$.  Show that a
sequence converges with respect to one of these metrics if and
only if it converges with respect to all of them.
(Hint: it's easier to compare the squares of $\rho_1,\rho_2,\rho_{max}$
than it is to compare the metrics themselves.)
\item[(iii)] Show that a sequence 
$\{\mathbf{x}_j\}_{j=1}^\infty\subset \mathbf{R}^n$
converges in the Euclidean metric if and only if it converges
``coordinate-wise''---i.e. when we write
$\mathbf{x}_j = (x_{j,1},\dots x_{j,n})$ out in terms of its
coordinates, then for each $k$ the sequence 
$\{x_{j,k}\}_{j=1}^\infty\subset\mathbf{R}$ converges.
 \end{description}

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