Triangle inequality

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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 in all inner product spaces; it is an axiom in the definition of abstract concepts such as normed vector spaces and metric spaces.

Normed vector space

In a normed vector space V, the triangle inequality reads

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

that is, the norm of the sum of two vectors is at most as large as the sum of the norms of the two vectors.

Metric space

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

that is, 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.

Consequences

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.