%\section{\defs} \textwidth=7in \evensidemargin=-.25in \oddsidemargin=-.25in \newtheorem{thm}{Theorem} %[section] \newtheorem{cor}[thm]{Corollary} \newtheorem{prop}[thm]{Proposition} \newtheorem{lem}[thm]{Lemma} \newtheorem{ex}[thm]{Exercise} \newtheorem{pres}{Presentation} \theoremstyle{definition} \newtheorem{defn}[thm]{Definition} \newtheorem{notn}[thm]{Notation} \newtheorem{eg}[thm]{Example} \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}