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, 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.