Norm

Definition · applies to vector spaces · also called $\norm{x}$, length, magnitude

Elementary

A vector is arrow between two points. Such a vector has a direction and a length. The norm of the vector is its length: how long the arrow is.

High school

Consider a vector $x = (x_1, \dots, x_n)$. The norm of $x$ is its length:

\[\norm{x} = \sqrt{x_1^2 + \dots + x_n^2}.\]

Caveat. This is the Euclidean norm, which is only one of many norms. Others include $\abs{x_1} + \dots + \abs{x_n}$ and $\max_i \abs{x_i}$.

Undergraduate

Consider a vector space $V$ and a function $\norm{\cdot} : V \to [0, \infty)$. The function is a norm on $V$ if

Examples

The Euclidean norm $x \mapsto \sqrt{x_1^2 + x_2^2}$ on $\reals^2$ High school
$x \mapsto \sqrt{x_1^2 + x_2^2}$ is a norm: the Euclidean norm, which comes from the dot product.
$x \mapsto \abs{x_1} + \abs{x_2}$ on $\reals^2$ Undergraduate
$x \mapsto \abs{x_1} + \abs{x_2}$ is a norm.

Things that may be mistaken for examples.

Great-circle distance on the surface of the Earth High school
Great-circle distance does not come from a norm, because the sphere is not a vector space.
$d(x, y) = 1$ for $x \neq y$, and $d(x, x) = 0$ (definition) Undergraduate
The discrete metric does not come from a norm, because it does not scale: $d(2x, 0) = 1 = d(x, 0)$ for every $x \neq 0$.

Every norm satisfies $\abs{\norm{x} - \norm{y}} \leq \norm{x - y}$, so it is a continuous function.
On a finite-dimensional space, all norms are equivalent: they give the same open sets and the same convergent sequences.

Norm vs. metric vs. inner product: Norms, metrics, and inner products measure size, distance, and angle, and each one gives the next in a natural way. A norm measures the lengths of vectors, a metric measures distances between points, and an inner product measures both lengths and angles. An inner product gives a norm, and a norm gives a metric, but not the reverse.