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 a metric space M, the triangle inequality is
- d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z in M
The following consequence of the triangle inequalities are often useful; they give lower bounds instead of upper bounds:
- | ||x|| - ||y|| | ≤ ||x + y||
- | d(x, y) - d(y, z) | ≤ d(x, z)
See also Cauchy-Schwarz inequality.