Sample LaTeX document of math homework, created by Robert Hanson.
Source: Robert Hanson webpage.
This template was originally published on ShareLaTeX and subsequently moved to Overleaf in October 2019.
\begin
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\documentclass[]{book}
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\usepackage{amsxtra}
\usepackage{amsthm}
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%Here I define some theorem styles and shortcut commands for symbols I use often
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{thm}{Theorem}
\newtheorem{cor}{Corollary}
\newtheorem*{rmk}{Remark}
\newtheorem{lem}{Lemma}
\newtheorem*{joke}{Joke}
\newtheorem{ex}{Example}
\newtheorem*{soln}{Solution}
\newtheorem{prop}{Proposition}
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\begin{document}
\begin{center}
{\Large Math 3220-1 \hspace{0.5cm} HW 1}\\
\textbf{NAME}\\ %You should put your name here
Due: DATE %You should write the date here.
\end{center}
\vspace{0.2 cm}
\subsection*{Exercises for Section 1.1: Norm and Inner Product}
\begin{enumerate}
\item\label{norms}
Define the \textdf{$\ell^1$-norm} on $\R^n$ by
$$\|x\|_1 = \sum_{i=1}^n |x^i|,$$
and define the \textdf{sup-norm} on $\R^n$ by
$$\|x\|_\infty = \sup\left\{|x^i|\right\}.$$
Show that these satisfy Theorem~1.
\begin{proof}
% WRITE YOUR PROOF HERE.
\end{proof}
\item Prove that $\ds \|x\|\leq \sum_{i=1}^n |x^i|$. In other words, the usual norm is no greater than the $\ell^1$-norm.
\begin{proof}
% WRITE YOUR PROOF HERE.
\end{proof}
\item Prove that $\|x-y\| \leq \|x\| + \|y\|$. (Compare this with part (2) of
Theorem~1.) When does equality hold?
\item Prove that $\ds \bigg| \, \|x\|-\|y\| \, \bigg| \leq \|x-y\|$.
\item The quantity $\|y-x\|$ is called the \textdf{distance} between $x$ and
$y$. Prove and interpret the ``triangle inequality'':
$$\|z-x\| \leq \|z-y\| + \|y-x\|.$$
\item\label{caushw} Let $f$ and $g$ be integrable on $[a,b]$.
\begin{enumerate}
\item Prove the integral version of the Cauchy-Schwarz inequality:
$$\left|\int_a^b fg\right| \leq \left(\int_a^b
f^2\right)^{1/2}\left(\int_a^b g^2\right)^{1/2}.$$
Hint: Consider separately the cases $0 = \int_a^b(f-t g)^2$ for
some $t\in\R$, and $0<\int_a^b(f-t g)^2$ for all
$t\in\R$.
\item If equality holds, must $f=t g$ for some $t\in\R$?
What if $f$ and $g$ are continuous?
\item Show that the Cauchy-Schwarz inequality is a special case of
(a).
\end{enumerate}
\item A linear transformation $T:\R^n\lra\R^n$ is \textdf{norm preserving} if
$$\|T(x)\|=\|x\|,$$ for all $x\in\R^n$, and \textdf{inner product preserving} if
$$\ip{Tx}{Ty} = \ip xy,$$ for all $x,y\in\R^n$.
\begin{enumerate}
\item Prove that $T$ is norm preserving if and only if it is inner
product preserving.
\item Prove that such a linear transformation is 1-1, and $T^{-1}$ is
norm preserving (and inner product preserving).
\end{enumerate}
\item\label{bddlin} If $T:\R^m\lra\R^n$ is a linear transformation, show that there is a
number $M$ such that $\|T(h)\|\leq M\|h\|$ for all $h\in\R^m$. Hint: Estimate
$\|T(h)\|$ in terms of $\|h\|$ and the entries in the matrix for $T$.
\item If $x,y\in\R^n$, and $z,w\in\R^m$, show that $\ds\ip{(x,z)}{(y,w)} = \ip xy
+ \ip zw$, and $\ds\|(x,z)\| = \sqrt{\|x\|^2 + \|z\|^2}$. Note that $(x,z)$ and
$(y,w)$ denote points in $\R^{n+m}$.
\item If $x,y\in\R^n$, then $x$ and $y$ are called \textdf{perpendicular} (or
\textdf{orthogonal}), and we write $x\perp y$, if $\ip xy = 0$. If $x\perp y$,
prove that $\|x+y\|^2 = \|x\|^2 + \|y\|^2$.
\end{enumerate}
\subsection*{Exercises for Section~1.2: More Topology: Open and Closed Sets in
$\R^n$}
\begin{enumerate}
\item\label{topology} Prove that the union of any (even infinite) number of open sets is open.
Prove that the intersection of two (and hence of finitely many) open sets is
open. Give a counterexample for the intersection of infinitely many open sets.
\item\label{clAB}If $A\subset B\subset\R^n$, prove that
$$\cl{A}\subset\cl{B},\quad\mbox{ and }\quad\inter{A}\subset\inter{B}.$$
\item Prove that if $B$ is an open subset of $A$, then $B\subset \inter(A)$. Note that this says that $\inter(A)$ is the largest open subset of $A$.
\item \label{openset}Prove that the $n$-dimensional ball centered at $a$ of radius $r$,
$$\ds B^n(a;r) = \left\{x\in\R^n:\|x-a\|< r\right\}$$ is open.
\item Find the interior, exterior, and boundary of the sets:
$$B^n = \left\{x\in\R^n : \|x\| \leq 1\right\},$$
$$S^{n-1} = \left\{x\in\R^n : \|x\| = 1\right\},$$
$$\Q^n = \left\{x\in\R^n : \mbox{ each } x^i\mbox{ is rational}\right\}.$$
\begin{soln}
% Put your answers here.
\end{soln}
\item If $A\subset[0,1]$ is the union of open intervals $(a_i,b_i)$ such that
each rational number in $(0,1)$ is contained in some $(a_i,b_i)$, show that
$\bd A = [0,1] - A$.
\item If $A$ is a closed set that contains every rational number $r\in[0,1]$,
show that $[0,1]\subset A$.
\item Graph generic open balls in $\R^2$ with respect to each of the ``non-Euclidean'' norms, $\|\cdot\|_1$ and $\|\cdot\|_\infty$.
What shapes are they?
\begin{soln}
% Put your figure and your write-up here. See the website on how to
% add a figure.
\end{soln}
\end{enumerate}
\end{document}