In mathematics, two metrics on the same underlying set are said to be equivalent if the resulting metric spaces share certain properties. Equivalence is a weaker notion than isometry; equivalent metrics do not have to be literally the same. Instead, it is one of several ways of generalizing equivalence of norms to general metric spaces.
Throughout the article, will denote a non-empty set and and will denote two metrics on .
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mathematics, two metrics on the same underlying set are said to be equivalent if the resulting metric spaces share certain properties. Equivalence is a weaker...
between seminorms and pseudometrics. Among examples ofmetrics induced by a norm are the metrics d1, d2, and d∞ on R 2 {\displaystyle \mathbb {R} ^{2}}...
ofequivalence classes is a metric space, the completion of M. The original space is embedded in this space via the identification of an element x of...
detail: The complex structure of the Riemann surface does uniquely determine a metric up to conformal equivalence. (Two metrics are said to be conformally...
The equivalence principle is the hypothesis that the observed equivalenceof gravitational and inertial mass is a consequence of nature. The weak form...
numbers can be formally defined as equivalence classes of pairs of integers (p, q) with q ≠ 0, using the equivalence relation defined as follows: ( p 1...
pseudometric induces an equivalence relation, called the metric identification, that converts the pseudometric space into a full-fledged metric space. This is...
e. a linear transformation of) the planar Manhattan distance. However, this geometric equivalence between L1 and L∞ metrics does not generalize to higher...
an equivalence relation on the set of geodesic rays, and the set ofequivalence classes is called the ideal boundary ∂X of X. An equivalence class of geodesic...
space onto itself. Being "homeomorphic" is an equivalence relation on topological spaces. Its equivalence classes are called homeomorphism classes. The...
mathematics, a ternary equivalence relation is a kind of ternary relation analogous to a binary equivalence relation. A ternary equivalence relation is symmetric...
Metric theories satisfy the Einstein Equivalence Principle. Extremely few non-metric theories satisfy this. For example, the non-metric theory of Belinfante...
always converge to a point Equivalenceofmetrics F-space – Topological vector space with a complete translation-invariant metric Fréchet space – A locally...
complete metrics and incomplete metrics generating their topology. There is a characterization of complete separable metric spaces in terms of a game known...
aforementioned equivalenceofmetric functions remains valid if √q(x − y) is replaced with M(x − y), where M is any convex positive homogeneous function of degree...
Income inequality metrics or income distribution metrics are used by social scientists to measure the distribution of income and economic inequality among...
the system. Because of the mass–energy equivalence this energy also has a mass-equivalent, which adds to the total mass–energy of the system, M {\displaystyle...
factor. An equivalence class of such metrics is known as a conformal metric or conformal class. Thus, a conformal metric may be regarded as a metric that is...
S2CID 121411457. Givens, Clark R; Shortt, Rae Michael (1984). "A class of Wasserstein metrics for probability distributions". Michigan Mathematical Journal. 31...
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