Metric

Definition

A metric on a set $X$ is a function $d: X \times X \to \mathbb{R}$ satisfying:

1. Positive definiteness: $d(x, y) \geq 0$, with equality iff $x = y$
2. Symmetry: $d(x, y) = d(y, x)$
3. Triangle inequality: $d(x, z) \leq d(x, y) + d(y, z)$

A set equipped with a metric is called a metric space.

From Norms

Every norm induces a metric:

$$ d(x, y) = \|x - y\| $$

Common Metrics

Metric	Formula	Name
$d(x,y) = \sqrt{\sum (x_i - y_i)^2}$	Euclidean distance	$L^2$ metric
$d(x,y) = \sum \|x_i - y_i\|$	Manhattan distance	$L^1$ metric
$d(x,y) = \max \|x_i - y_i\|$	Chebyshev distance	$L^\infty$ metric
$d(x,y) = \begin{cases} 0 & x = y \\ 1 & x \neq y \end{cases}$	Discrete metric

Intuition

A metric generalizes the concept of “distance” to abstract spaces. It tells you how far apart two points are, satisfying our intuitive expectations:

• Distance is non-negative
• Distance from $x$ to $y$ equals distance from $y$ to $x$
• Going through an intermediate point can’t be shorter than going directly

Related Concepts

• Norm — induces a metric via $d(x,y) = \|x - y\|$
• metric_space — a set with a metric
• continuity — defined using metrics