\documentclass{amsart}
\newcommand{\F}{\mathbb F}
\newcommand{\vv}{\vec{v}}
\newcommand{\vu}{\vec{u}}
\renewcommand{\L}{\mathcal L}
\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\bilin}[1]{\langle#1\rangle}

\theoremstyle{definition}
\newtheorem{defn}{Definition}
\theoremstyle{plain}
\newtheorem{thm}{Theorem}
\newtheorem{prop}{Proposition}
\newtheorem{lem}{Lemma}
\newtheorem{cor}{Corollary}

\title{Lecture 23: Metric Spaces}

\begin{document}
\maketitle

This lecture will look at the topological properties that arise from the notion of distance provided by the norm.

\begin{defn}
If $V$ is an inner product space, then the distance between $\vv$ and $\vu$ is
\[
d(\vv,\vu)=||\vv-\vu||.
\]
\end{defn}

\begin{prop}
The distance function satisfies
\begin{enumerate}
\item $d(\vv,\vu)\geq 0$ with equality iff $\vv=\vu$.
\item $d(\vv,\vu)=d(\vu,\vv)$ for all $\vu$ and $\vv$
\item $d(\vv,\vu)=d(\vv,\vec{w})+d(\vec{w},\vu)$ for all $\vv$, $\vu$, $\vec{w}$.
\end{enumerate}
\end{prop}
\begin{proof}
The first condition is positive-definteness, the second is symmetry, and the third is the triangle inequality.
\end{proof}

\begin{defn}
A metric space is a set $M$ with a function $d\colon M\times M\to\R$ that satisfies the three conditions in the previous proposition.
\end{defn}

Today's lecture will focus on metric spaces and establish some of the basic terminology. Next time we will start to look at the algebraic properties.

First some specific terms. 

\begin{defn}
The open ball of radius $r$ about $x$ is 
\[
B_r(x)=\{y\in M | d(x,y)<r\}.
\]

The closed ball of radius $r$ about $x$ is
\[
\overline{B_r(x)}=\{y\in M | d(x,y)\leq r\}.
\]
\end{defn}

The names come from $\R^2$ with the standard distance. Here
\[
B_r(x)=\{(x,y) \in\R^2 | d\big((a,b),(x,y)\big)<r\}=\{(x,y) | (x-a)^2+(y-b)^2<r^2\}.
\]

We can, however, put other distances on $\R^n$.

\begin{defn}
A \underline{norm} on $V$ is a function $||-||\colon V\to\R$ such that
\begin{enumerate}
\item $||\vv||\geq 0$ with equality iff $\vv=\vec{0}$.
\item $||r\vv||=|r|\cdot ||\vv||$
\item $||\vv+\vu||\leq ||\vv||+||\vu||$.
\end{enumerate}
\end{defn}

There are a number of important examples.

\begin{enumerate}
\item Let $p\geq 1$, and let $||-||_p$ on $\R^n$ be
    \[
    ||\vv||_p=\sqrt[p]{\sum |a_i|^p}.
    \]
    If $p=2$, then this is our Euclidean distance. If $p=1$, then this is the so called ``taxi-cab'' distance, since the distance between two points on the plane is how far a taxi would drive if it could only travel parallel to the x or y axis. For other values of $p$, we don't have such a fun description.
    
    The unit ball centered at the origin in each of these looks very different depending on the value of $p$. If $p=1$, then it looks like a diamond with vertices at $(\pm 1,0)$ and $(0,\pm 1)$. If $p=2$, then it is the standard unit circle, and as $p\to\infty$, it grows to the square with vertices at $(\pm 1,\pm 1)$.
\item Let $||-||_\infty$ on $\R^n$ be
\[
||\vv||_\infty=\max_{i\leq n} |a_i|.
\]
This serves as a kind of limit of the $||-||_p$ (if I hold $\vv$ constant and let $p\to\infty$, then the norms converge to the infinity norm).

The unit ball in this metric space is the square with vertices at $(\pm 1, \pm 1)$.

\item $\ell_p$: these are all infinite sequences of real numbers such that the $p$-norm makes sense (in other words, such that the associated sequence converges). For the infinity norm, these are all bounded sequences.
    
\item $L^p([a,b])$: these are all functions $[a,b]\to\R$ such that $\int_a^b |f(x)|^p dx <\infty$. The norm is then the $p$\textsuperscript{th} root of this integral. In particular, all continuous functions sit inside this space, but there are also discontinuous functions here too.
\end{enumerate}

The reason we consider metric spaces is that we can use this to talk about convergences of sequences.

\begin{defn}
Let $(x_n)$ be a sequence of elements in a metric space $M$. We say that $(x_n)$ \underline{converges} to $x$ if for every $\epsilon >0$, we can find an $N$ such that for all $n>N$, $x_n\in B_\epsilon(x)$.
\end{defn}
\end{document} 