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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, Lp 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, 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 consequence of the triangle inequalities are often useful; they give lower bounds instead of upper bounds:

| ||x|| - ||y|| | ≤ ||x + y||
which expresses the fact that the norm is a continuous map, and
| d(x, y) - d(y, z) | ≤ d(x, z)
which says that the metric is a continuous map.

See also Cauchy-Schwarz inequality.