Section 1
The Definition and Some Examples
For $a=(-2,1)$ and $b=(3,4)$ in $R^2$, compute the distance from $a$ to $b$ in each of the following metrics: (a) usual, (b) taxicab, (c) $\max$, (d) discrete.
Determine the distance from $(3,4)$ to the unit square $[0,1] \times[0,1]$ in $R^2$ with respect to each of the four metrics listed in Problem 1.
Prove that the taxicab metric $d^n$ is actually a metric for $\mathbb{R}^n$.
Prove that each of the following functions is a metric:(a) the discrete metric of Example 3.1.3(b) the function $\rho$ of Example 3.1.4(c) the function $\rho^{\prime}$ of Example 3.1.5
Describe pictorially in $\mathbb{R}^2$ the set of points $x$ whose distance from the origin is less than or equal to 1 with respect to each of the following metrics: (a) usual, (b) taxicab, (c) max, (d) discrete.
Repeat Problem 5 for the set of points whose distance from the origin is less than 1.
Describe pictorially (on a graph) the set of functions $g$ in $e[a, b]$ whose distance from a given function $f$ is less than or equal to 1 for each of the following metrics: (a) the integral metric $\rho$ of Example 3.1.4, (b) the supremum metric $\rho^{\prime}$ of Example 3.1.5, (c) the discrete metric.
Let $B=\left\{x=\left(x_1, x_2, x_3\right) \in \mathbb{R}^3: x_1^2+x_2^2+x_3^2 \leq 1\right\}$ be the unit ball in $\mathbb{R}^3$. Compute the diameter of $B$ for each of the following metrics: (a) usual, (b) taxicab, (c) max, (d) discréte.
Show that if $(X, d)$ is a metric space with discrete metric $d$ and $A$ is a subset of $X$ with at least two members, then the diameter of $A$ is 1 .
Let $A=\left\{x=\left(x_1, x_2\right) \in R^2: x_1^2+x_2^2 \leq 1\right\}$ and let $b=(1,1)$. Find the distance from $b$ to $A$ for the following metrics: (a) usual, (b) taxicab, (c) $\max$, (d) discrete.
Let $(X, d)$ be a metric space and $A$ a subset of $X$. Prove that the diameter of $A$ is zero if and only if $A$ has fewer than two members.