In mathematics, triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Look up theorem in Wiktionary, the free dictionary. ... A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line segments. ...

The triangle inequality is a theorem in spaces such as the real numbers, all Euclidean spaces, the L^p spaces (p ≥ 1), and any inner product space. It also appears as an axiom in the definition of many structures in mathematical analysis and functional analysis, such as normed vector spaces and metric spaces. In mathematics, the real numbers may be described informally in several different ways. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...

Normed vector space

In a normed vector space V, the triangle inequality is In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ...

||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.

The real line is a normed vector space with the absolute value as the norm, and so the triangle inequality states that for any real numbers x and y: In mathematics, the real line is simply the set of real numbers. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...

The triangle inequality is useful in mathematical analysis for determining the best upper estimate on the size of the sum of two numbers, in terms of the sizes of the individual numbers. Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ...

There is also a lower estimate, which can be found using the inverse triangle inequality which states that for any real numbers x and y:

and

If the norm arises from an inner product (as is the case for Euclidean spaces), then the triangle inequality follows from the Cauchy–Schwarz inequality. In mathematics, the Cauchy–Schwarz inequality, also known as the Schwarz inequality, the Cauchy inequality, or the Cauchy–Bunyakovski–Schwarz inequality, named after Augustin Louis Cauchy, Viktor Yakovlevich Bunyakovsky and Hermann Amandus Schwarz, is a useful inequality encountered in many different settings, such as linear algebra applied to vectors, in...

Metric space

In a metric space M with metric d, the triangle inequality is In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...

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)

| ||x|| − ||y|| | ≤ ||x + y||

this implies that the norm ||–|| as well as the distance function d(x, –) are 1-Lipschitz and therefore continuous. In mathematics, a function f : D → R defined on a set D of real numbers with real values is called Lipschitz continuous (or is said to satisfy a Lipschitz condition) if there exists a constant K ≥ 0 such that for all in D. The smallest such K is called the... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

Reversal in Minkowski space

In the usual Minkowski space and in Minkowski space extended to an arbitrary number of spatial dimensions, assuming null or timelike vectors in the same time direction, the triangle inequality is reversed: In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...

||x + y|| ≥ ||x|| + ||y||     for all x, y in V such that ||x|| ≥ 0, ||y|| ≥ 0 and t_x t_y ≥ 0

A physical example of this inequality is the twin paradox in special relativity. This article or section is in need of attention from an expert on the subject. ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest...

See also

• Subadditivity

A function f(x) is subadditive if for all x and y in the domain of f. ...

External link

• Triangle inequality demonstration With interactive animation