\documentclass[11pt]{article}
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{epstopdf}
\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png}

\textwidth = 6.5 in
\textheight = 9 in
\oddsidemargin = 0.0 in
\evensidemargin = 0.0 in
\topmargin = 0.0 in
\headheight = 0.0 in
\headsep = 0.0 in
\parskip = 0.2in
\parindent = 0.0in

\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}{Definition}

\begin{document}

\begincenter{}\endcenter







\centerline{\bf Math 366, Final Exam}
\vskip2mm

This is a 100 point exam. 
\smallskip 


1. (20 points) 
(a) State PRECISELY the Arzela-Ascoli theorem. 
\vskip45mm 

(b) State PRECISELY the Inverse Function Theorem. 
\vskip45mm 

(c) Define contraction mapping and state the  contraction mapping theorem. 
\vskip45mm

\vfill\eject

2. (15 points) Let $M_{mn}$ denote the collection of all $m\times n$ matrices. 

(1) Define two different norms in $M_{mn}$ 

(2) Compare these two norms for any element in $M_{mn}$. 
\vfill\eject
 

3. (20 points) Let $X$ be a metric space and $M$ a subset in $X$. 

(1) Define precisely that $M$ is compact in $X$. 

(2) Prove by the definition that if $M$ is compact then $M$ is closed. 

(3) Suppose $M$ is compact and $F\subset M$ is closed. Prove that 
$F$ is compact. 

\vfill\eject

4. (10 points) Suppose that $X$ is metric space and that 
$\{f_n\}$ is uniformly convergent sequence of bounded 
functions on $X$. Is it true that $\{f_n\}$ is uniformly bounded? 
Prove or give a counter example. 
\vfill\eject

 
5. (15 points) Consider 
$$\eqalign  
{f_1(x_1, x_2, x_3, y_1, y_2)&=e^{y_1}+x_2y_2-9x_1x_3-1=0\cr   
f_2(x_1, x_2, x_3, y_1, y_2)&=5y_2g(y_1)+g(y_1+2)+2x_2+8x_3=0, 
\cr} 
$$ 
where $g(y_1)$ is continuously differentiable. We want to solve 
$(y_1, y_2)$ in terms of $(x_1, x_2, x_3)$ near $(0, 0, 0, 0,0)$. 
Find the condition(s) on $g(y_1)$ so that the implicit function theorem 
can be used near $(0, 0, 0, 0,0)$. 
 
\vfill\eject

6. (20 points) Consider the family of continuous functions 
$$f_n(x, y)={{x^2+y^2+5xy}\over{x^2+y^2+({1\over n})^2}}$$ 
on the disk $B_1=\{(x, y); x^2+y^2\le 1\}$.  

(1) Find the pointwise limit $f(x, y)$ of $f_n(x, y)$ on $B_1$. 

(2) Show that $f$ is not continuous. 

(3) Is the convergence $f_n(x, y)\to f(x, y)$ uniform on $B_1$? 
Justify your result. 

\vfill\eject

\end{document} 
