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Concept
Pseudometric space
Summary
In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by Đuro Kurepa in 1934. In the same way as every normed space is a metric space, every seminormed space is a pseudometric space. Because of this analogy the term semimetric space (which has a different meaning in topology) is sometimes used as a synonym, especially in functional analysis.
When a topology is generated using a family of pseudometrics, the space is called a gauge space.
A pseudometric space is a set together with a non-negative real-valued function called a , such that for every
Symmetry:
Subadditivity/Triangle inequality:
Unlike a metric space, points in a pseudometric space need not be distinguishable; that is, one may have for distinct values
Any metric space is a pseudometric space.
Pseudometrics arise naturally in functional analysis. Consider the space of real-valued functions together with a special point This point then induces a pseudometric on the space of functions, given by for
A seminorm induces the pseudometric . This is a convex function of an affine function of (in particular, a translation), and therefore convex in . (Likewise for .)
Conversely, a homogeneous, translation-invariant pseudometric induces a seminorm.
Pseudometrics also arise in the theory of hyperbolic complex manifolds: see Kobayashi metric.
Every measure space can be viewed as a complete pseudometric space by defining for all where the triangle denotes symmetric difference.
If is a function and d2 is a pseudometric on X2, then gives a pseudometric on X1. If d2 is a metric and f is injective, then d1 is a metric.
The is the topology generated by the open balls
which form a basis for the topology. A topological space is said to be a if the space can be given a pseudometric such that the pseudometric topology coincides with the given topology on the space.
The difference between pseudometrics and metrics is entirely topological.
Official source
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