%\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} \theoremstyle{definition} \newtheorem{defn}[thm]{Definition} \newtheorem{notn}[thm]{Notation} \newtheorem{eg}[thm]{Example} \newcommand{\C}{\Bbb C} \newcommand{\Z}{\Bbb Z} \newcommand{\F}{\Bbb F} \newcommand{\R}{\Bbb R} \newcommand{\Q}{\Bbb Q} \newcommand{\quats}{\Bbb H} \newcommand{\GL}{\operatorname{GL}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\Span}{\operatorname{Span}} \newcommand{\stab}{\operatorname{Stab}} \newcommand{\cl}{\operatorname{cl}} \newcommand{\normal}{\triangleleft} \begin{document} \title{Math 101--Analysis Seminar} \maketitle \section*{A preliminary note} Dear Analysts, \bigskip In a previous message I wrote: \begin{quotation} Some time before next semester begins we have to prepare for our first meeting. (Otherwise we won't have enough to talk about for a full meeting.) I will send out a more formal assignment during break, but roughly speaking we will start by discussing Chapter 1 in the text. (Analysis on Manifolds by Munkres) This should be mostly review for most of you. Look at this chapter and the associated problems over break. If you find anything which looks totally alien, let me know if you can so I can adjust my expectations. \end{quotation} Here, I would like to make some more specific suggestions. I understand that not all of you will be able to do all of this, but if all of you do what you can, we will have enough to talk about for our first meeting. I will hand out a more formal syllabus at our first meeting. Throughout, I will make a distinction between ``doing'' problems and ``writing'' problems. I expect that you will write down some sort of solution for your own records when I ask you to do a problem. I don't expect this record to be necessarily expertly written, but I do expect it to be complete in the sense that little or no mathematical thought need be added in the process of writing up the solution expertly. When I ask you to write a problem, I will want a polished and careful writing suitable for your fellow students (and your professor) to read. \section*{Reading} Read sections one through four in Munkres. \section*{Doing} As you read Section 1 be sure to ask yourself whether you have some idea of how to prove each of the unproved assertions. Assertion 1.3 will occur as exercise 1.2. \begin{ex} Do exercises 1.1 through 1.4. Look at exercise 1.5 and do it if you have time. \end{ex} \begin{ex} Look at all the problems in section 2. Do the first one and the last one and any of the others which don't look familiar. \end{ex} \begin{ex} Show that a square matrix with integer entries is invertible if and only if it has determinant \( \pm 1 \). \end{ex} \begin{ex} Look at all the problems in section 3. Do 3.6 \end{ex} Read section four carefully. \begin{ex} In section 4 (and afterwards) Munkres occasionally uses the sup norm and occasionally uses the Euclidean norm. In section 4, find all instances of each and try to explain to yourself why he uses which norm. \end{ex} \begin{ex} Do problems 4.1, 4.2, 4.3, and 4.4. \end{ex} \section*{Writing} Write up one or more of the problems you have done and hand it in to me so that I can critique the result. Altogether, I would like a page or two of carefully written exposition. If possible, pick from among the problems that you did not know how to do before you did them. \end{document}