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Definition. Formally, a metric space is an ordered pair (M, d) where M is a set and d is a metric on M, i.e., a function satisfying the following axioms for all points : [4][5] The distance from a point to itself is zero: (Positivity) The distance between two distinct points is always positive:
Complete metric space. In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because ...
Space (mathematics) In mathematics, a space is a set (sometimes known as a universe) endowed with a structure defining the relationships among the elements of the set. A subspace is a subset of the parent space which retains the same structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces ...
Metric map. In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance. These maps are the morphisms in the category of metric spaces, Met. [1] Such functions are always continuous functions. They are also called Lipschitz functions with Lipschitz constant 1, nonexpansive ...
A metric space is a length metric space if the intrinsic metric agrees with the original metric of the space. If the space has the stronger property that there always exists a path that achieves the infimum of length (a geodesic) then it is called a geodesic metric space or geodesic space. For instance, the Euclidean plane is a geodesic space ...
Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space.
The normed vector space ((,), ‖ ‖) is called space or the Lebesgue space of -th power integrable functions and it is a Banach space for every (meaning that it is a complete metric space, a result that is sometimes called the Riesz–Fischer theorem).
In mathematics, probabilistic metric spaces are a generalization of metric spaces where the distance no longer takes values in the non-negative real numbers R ≥ 0, but in distribution functions. [1] Let D+ be the set of all probability distribution functions F such that F (0) = 0 (F is a nondecreasing, left continuous mapping from R into [0 ...