Norm (mathematics)

In mathematics, the norm of a vector is its length. For the real numbers the only norm is the absolute value. For spaces with more dimensions the norm can be any function $p$ with

1. Scales for real numbers $a$, that is $p(ax) = |a|p(x)$
2. Function of sum is less than sum of functions, that is $p(x + y) \leq p(x) + p(y)$ or the triangle inequality
3. $p(x) = 0$ if and only if $x = 0$.

Examples

1. The one-norm is the sum of absolute values: $\|x\|_1 = |x_1| + |x_2| + ... + |x_N|.$ This is like finding the distance from one place on a grid to another by summing together the distances in all directions the grid goes; see Manhattan Distance
2. Euclidean norm is the sum of the squares of the values: $\|x\|_2 = \sqrt{x_1^2 + x_2^2 + ... + x_N^2}$
3. Maximum norm is the maximum absolute value: $\|x\|_{\infty} = \max(|x_1|,|x_2|,...,|x_N|)$