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 [math]\displaystyle{ p }[/math] with

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

Examples

1. The one-norm is the sum of absolute values: [math]\displaystyle{ \|x\|_1 = |x_1| + |x_2| + ... + |x_N|. }[/math] 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: [math]\displaystyle{ \|x\|_2 = \sqrt{x_1^2 + x_2^2 + ... + x_N^2} }[/math]
3. Maximum norm is the maximum absolute value: [math]\displaystyle{ \|x\|_{\infty} = \max(|x_1|,|x_2|,...,|x_N|) }[/math]

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