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Metric mathematics - Definition | | Metric mathematics - Definition: Encyclopedia II - Metric mathematics - Definition | | A metric on a set X is a function (called the distance function or simply distance)
d : X × X → R
(where R is the set of real numbers). For all x, y, z in X, this function is required to satisfy the following conditions:
d(x, y) ≥ 0 (non-negativity)
d(x, y) = 0 if and only if x = y (identity ...
See also:Metric mathematics, Metric mathematics - Definition, Metric mathematics - Notes, Metric mathematics - Examples, Metric mathematics - Equivalence of metrics, Metric mathematics - Relation of norms and metrics, Metric mathematics - Related concepts and alternative axiom systems | | | Metric mathematics, Metric mathematics - Definition, Metric mathematics - Equivalence of metrics, Metric mathematics - Examples, Metric mathematics - Notes, Metric mathematics - Related concepts and alternative axiom systems, Metric mathematics - Relation of norms and metrics, Distance, Metric space, Metric tensor | | |
| | | Metric mathematics: Encyclopedia II - Metric mathematics - Definition
Metric mathematics - Definition
A metric on a set X is a function (called the distance function or simply distance)
d : X × X → R
(where R is the set of real numbers). For all x, y, z in X, this function is required to satisfy the following conditions:
- d(x, y) ≥ 0 (non-negativity)
- d(x, y) = 0 if and only if x = y (identity of indiscernibles)
- d(x, y) = d(y, x) (symmetry)
- d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality).
A metric d on X is called intrinsic if any two points x and y in X can be joined by a curve with length arbitrarily close to d(x, y).
For sets on which an addition + : X × X → X is defined, we call d a translation invariant metric if
d(x, y)=d(x + a, y + a)
for all x,y and a in X.
If the triangular inequality is strengthened to
d(x, z) ≤ max( d(x, y), d(y, z) )
the metric is called ultrametric, see below.
Other related archivesApproach spaces, Distance, Euclid, Euclidean metric, Metric geometry, Metric space, Metric tensor, Pseudo-Riemannian metric tensor, Riemannian manifold, Topology, categorical, continuity, convergence, curve, differentiability, differential geometry, directed sets, discrete metric, distance, extended real number line, function, homeomorphism, inner products, integration, intrinsic, length, locally convex, manifold, mathematics, metric, metric space, metric tensors, metrisable, norm, normed vector space, positive, pseudometric spaces, quasimetric spaces, real numbers, seminorm, seminorms, sequence, set, summable sequence, tangent space, theory of relativity, topological vector space, topology, triangle inequality, ultrametric, uniform isomorphism, uniform spaces, vector space, work
 Adapted from the Wikipedia article "Definition", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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