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\newtheorem{thm}{Theorem} %[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{prop}[thm]{Proposition}
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\newcommand{\R}{{\Bbb R}}
\newcommand{\GL}{\operatorname{GL}}
\begin{document}
\title{Analysis Seminar: Assignment 2}
\maketitle
\section*{Reading}
Read sections eight and nine in Munkres.  Usually, we will cover more 
than two sections, but in this case I think the material is dense 
enough that two will keep us busy.  Here are some questions you will 
want to ask yourself while doing the reading. This list is certainly not 
exhaustive, but  if these were not already obvious to you, 
I hope they will be some help.
\begin{itemize}
	\item  How could anyone known to have used the mean value theorem to 
	prove 8.1?   

	\item  How could anyone have arrived at this proof of 8.1 without 
	reading the proof somewhere?

	\item  Where is the hypothesis that \( f \) is one-to-one used in the 
	proof of 8.2?

	\item  In the middle of page 74 we ``conclude'' that there is an open 
	set \( U\times V \), however not all open sets are of this form.  Why 
	can we assume there is an open set of this form here?

	\item  Why are there straight sides on the picture at the top of page 
	75?
\end{itemize}
\section*{Doing}
Do all eleven problems in this section.  Also do the following 
exercise (not too hard) which may or may not help you to think about 
what is going on in step five of Theorem 8.2.
\begin{ex}
	Show that \( \GL(n) \) is an open subset of \( \R^{n^{2}} \).
\end{ex}
\section*{Writing}
Write up 8.2, 8.5, 9.2, and 9.6 expertly.
\section*{Presentations}
I have assigned two people to each presentation.  You may split up the 
work however the two of you see fit, as long as both of you speak for 
part of the time when it comes time to present.  Try to have an idea 
of the kind of thing you would like to do by the time you come to see 
me on Friday.  Sign up for a pair of adjacent slots and come together.
\begin{pres}[Aaron, Tom]
	The inverse function theorem.
\end{pres}
\begin{pres}[Peter, Rob]
    The implicit function theorem.
\end{pres}

\end{document}