A norm assigns a nonnegative size to a vector. The Euclidean norm is

$$\lVert x\rVert_2=\sqrt{x_1^2+\cdots+x_n^2}.$$

It generalizes ordinary geometric length.

Other useful norms include

$$\lVert x\rVert_1=\sum_i|x_i|$$

and

$$\lVert x\rVert_\infty=\max_i|x_i|.$$

A norm satisfies three essential properties: it is zero only for the zero vector, scaling a vector by $a$ scales its norm by $|a|$, and

$$\lVert x+y\rVert\le\lVert x\rVert+\lVert y\rVert.$$

A norm induces a distance between vectors:

$$d(x,y)=\lVert x-y\rVert.$$

Different norms measure size differently, so statements such as “small error” or “nearest point” are incomplete until the relevant notion of distance is clear.