Triangle inequality

In mathematics, the triangle inequality is a statement which states roughly that the distance from A to B to C is never shorter than going directly from A to C. The triangle inequality is a theorem in spaces such as the real numbers, Euclidean space, L^p spaces (p ≥ 1) and more generally in all inner product spaces; it is an axiom in the definition of abstract concepts such as normed vector spaces and metric spaces.

In a normed vector space V, the triangle inequality reads

||x + y|| ≤ ||x|| + ||y|| for all x, y in V

in words: "the norm of the sum of two vectors is at most as large as the sum of the norms of the two vectors."

In a metric space M with metric d, the triangle inequality is

d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z in M

in words: the distance from x to z is at most as large as the sum of the distance from x to y and the distance from y to z.

The following consequences of the triangle inequalities are often useful; they give lower bounds instead of upper bounds:

| ||x|| - ||y|| | ≤ ||x - y|| or for metric | d(x, y) - d(x, z) | ≤ d(y, z)

this implies that the norm ||-|| as well distance function d(x, -) are 1-Lipschitz and therefore continuous.