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\newtheorem{thm}{Theorem} %[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{ex}[thm]{Exercise}
\newtheorem{pres}{Presentation}

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\newcommand{\R}{{\Bbb R}}
\newcommand{\GL}{\operatorname{GL}}
\begin{document}
\title{Analysis Seminar: Assignment 2}
\maketitle
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

\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
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