Norm (mathematics)

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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[change | edit source]

  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|)