norm (idea)

In mathematics, a generalization of the concept of absolute value. A norm || || on a vector space E over a field k is a way of measuring the distance of elements of E from zero. Usually norms are required to satisfy the following axioms:

• ||x|| is a nonnegative real number for every x ∈ E, and ||x|| = 0 if and only if x = 0;
• if α ∈ k and x ∈ E, then ||αx|| = |α| ||x||, where |α| is measured in the usual absolute value on k;
• if x, y ∈ E, then ||x + y|| <= ||x|| + ||y||. This is the familiar triangle inequality.

Weakening one or another of these axioms yields various generalizations such as quasinorms, pseudonorms, etc. A vector space endowed with a norm is called a normed linear space. See Banach space for more and examples.