Metric (mathematics)

In mathematics, a metric or distance function is a function which defines a distance between elements of a set. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Distance is a numerical description of how far apart objects are A set with a metric is called a metric space. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined A metric induces a topology on a set but not all topologies can be generated by a metric. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of When a topological space has a topology that can be described by a metric, we say that topological space is metrizable. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Topology and related areas of Mathematics, a metrizable space is a Topological space that is homeomorphic to a Metric space.

In differential geometry, the word "metric" is also used to refer to a structure defined only on a vector space which is more properly termed a metric tensor (or Riemannian or pseudo-Riemannian metric). Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space

1 Definition
2 Notes
3 Examples
4 Equivalence of metrics
5 Metrics on vector spaces
6 Generalized metrics
7 See also

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). The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, the real numbers may be described informally in several different ways 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. A negative number is a Number that is less than zero, such as −2 The identity of indiscernibles is an ontological principle which states that two or more objects or entities are identical (are one and the same entity Note that condition 1 and 2 together produce positive definiteness)
d(x, y) = d(y, x) (symmetry)
d(x, z) ≤ d(x, y) + d(y, z) (subadditivity / triangle inequality). In Mathematics, the term positive-definite function may refer to a couple of different concepts In Mathematics, a Binary relation R over a set X is symmetric if it holds for all a and b in X that In Mathematics, subadditivity is a property of a function that states roughly that evaluating the function for the sum of two elements of the Domain In Mathematics, the triangle inequality states that for any Triangle, the length of a given side must be less than or equal to the sum of the other two sides but greater

These axioms are not independent: Non-negativity follows from the other axioms.

A metric is called an ultrametric if it satisfies the following stronger version of the triangle inequality:

For all x, y, z in M, d(x, z) ≤ max(d(x, y), d(y, z))

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). In Mathematics, an ultrametric space is a special kind of Metric space in which the Triangle inequality is replaced with d ( x, In the mathematical study of Metric spaces, one can consider the Arclength of paths in the space In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object

For sets on which an addition + : X × X → X is defined, d is called a translation invariant metric if

d(x, y) = d(x + a, y + a)

for all x, y and a in X. In Geometry, a translation "slides" an object by a vector a: T a (p = p + a

Notes

These conditions express intuitive notions about the concept of distance. Distance is a numerical description of how far apart objects are For example, that the distance between distinct points is positive and the distance from x to y is the same as the distance from y to x. The triangle inequality means that the distance traversed directly between x and z, is not larger than the distance to traverse in going first from x to y, and then from y to z. Euclid in his work stated that the shortest distance between two points is a line; that was the triangle inequality for his geometry. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria.

Property 1 (d(x, y) ≥ 0) follows from properties 2 and 4 and does not have to be required separately.

Examples

The discrete metric: if x = y then d(x,y) = 0. In Topology, a discrete space is a particularly simple example of a Topological space or similar structure one in which the points are " isolated " Otherwise, d(x,y) = 1.
The Euclidean metric is translation and rotation invariant. In Mathematics, the Euclidean distance or Euclidean metric is the "ordinary" Distance between two points that one would measure with a ruler
The Manhattan metric is translation invariant. Taxicab geometry, considered by Hermann Minkowski in the 19th century is a form of Geometry in which the usual metric of Euclidean geometry
More generally, any metric induced by a norm (see below) is translation invariant. In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length
If (p_n)_n∈N is a sequence of seminorms defining a (locally convex) topological vector space E, then

$d(x,y)=\sum_{n=1}^\infty \frac{1}{2^n} \frac{p_n(x-y)}{1+p_n(x-y)}$

is a metric defining the same topology. In Mathematics, a sequence is an ordered list of objects (or events In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length In Functional analysis and related areas of Mathematics, locally convex topological vector spaces or locally convex spaces are examples of Topological In Mathematics, a topological vector space is one of the basic structures investigated in Functional analysis. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of (One can replace $\frac{1}{2^n}$ by any summable sequence

(a n)

of strictly positive numbers. In Mathematics, a series (or sometimes also an Integral) is said to converge absolutely if the sum (or integral of the Absolute value of the )

Equivalence of metrics

For a given set X, two metrics d₁ and d₂ are called topologically equivalent (uniformly equivalent) if the identity mapping

id: (X,d₁) → (X,d₂)

is a homeomorphism (uniform isomorphism). Topological equivalence redirects here see also Topological equivalence (dynamical systems. In the mathematical field of Topology a uniform isomorphism or uniform homeomorphism is a special Isomorphism between Uniform spaces

For example, if $d$ is a metric, then $min(d,1)$ and ${d \over 1+d}$ are metrics equivalent to $d .$

See also notions of metric space equivalence. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined

Metrics on vector spaces

Given a normed vector space (X,||. 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 ||) we can define a metric on X by

d(x,y):=||x-y||.

The metric d is said to be induced by the norm ||. ||.

Conversely if a metric d on a vector space X satisfies the properties

d(x,y) = d(x+a,y+a) (translation invariance)
d(αx,αy) = |α|d(x,y) (homogeneity)

then we can define a norm on X by

||x||:=d(x,0)

Similarly, a seminorm induces a pseudometric (see below), and a homogeneous, translation invariant pseudometric induces a seminorm. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length

Generalized metrics

Extending the range

Some authors allow the distance function d to attain the value ∞, i. e. distances are non-negative numbers on the extended real number line. In Mathematics, the affinely extended real number system is obtained from the Real number system R by adding two elements +∞ and &minus∞ (pronounced Such a metric is called an extended metric. Every extended metric can be transformed to a finite metric such that the metric spaces are equivalent as far as notions of topology (such as continuity or convergence) are concerned. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" This can be done using a subadditive monotically increasing bounded function which is zero at zero, e. In Mathematics, subadditivity is a property of a function that states roughly that evaluating the function for the sum of two elements of the Domain g. d'(x, y) = d(x, y) / (1 + d(x, y)) or d''(x, y) = min(1, d(x, y))).

The requirement that the metric takes values in [0,∞) can even be relaxed to consider metrics with values in other directed sets. In Mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive The reformulation of the axioms in this case leads to the construction of uniform spaces: topological spaces with an abstract structure enabling one to compare the local topologies of different points. In the Mathematical field of Topology, a uniform space is a set with a uniform structure.

Relaxing the axioms

If the second requirement (indiscernibility) is relaxed to the condition d(x,x)=0 for all x, the function is called a pseudometric. In Mathematics, a pseudometric space is a generalized Metric space in which the distance between two distinct points can be zero This is the most common generalization of metrics. In topology, a semimetric is a function that satisfies the first three axioms, but not necessarily the triangle inequality. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Topology, a semimetric space is a generalized Metric space in which the Triangle inequality is not required Occasionally, a quasimetric is defined as a function that satisfies all axioms for a metric with the possible exception of symmetry. In Mathematics, a quasimetric space is a generalized Metric space in which the metric is not necessarily symmetric Sometimes the presyllables are combined; the meaning is obvious.

These notions are not completely standardized. In particular, the term semimetric is often used as a synonym for pseudometric (especially in functional analysis). In Topology, a semimetric space is a generalized Metric space in which the Triangle inequality is not required In Mathematics, a pseudometric space is a generalized Metric space in which the distance between two distinct points can be zero For functional analysis as used in psychology see the Functional analysis (psychology article

An example of a distance function that does not satisfy the identity of indiscernibles condition 2. The identity of indiscernibles is an ontological principle which states that two or more objects or entities are identical (are one and the same entity is the probability metric. A probability metric is a function defining a distance between Random variables or vectors.

Important cases of generalized metrics

From a categorical point of view, the extended pseudometric and the extended pseudoquasimetric spaces, along with their corresponding nonexpansive maps, are the best behaved of the metric space categories. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets One can take arbitrary products and coproducts and form quotient objects within the given category. If one drops "extended", one can only take finite products and coproducts. If one drops "pseudo", one cannot take quotients. Approach spaces are a generalization of metric spaces that maintains these good categorical properties. In Topology, approach spaces are a generalization of Metric spaces based on point-to-set distances instead of point-to-point distances

In differential geometry, one considers metric tensors, which can be thought of as "infinitesimal" metric functions. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space They are defined as inner products on the tangent space with an appropriate differentiability requirement. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change While these are not metric functions as defined in this article, they induce metric functions by integration. In Calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose Derivative A manifold with a metric tensor is called a Riemannian manifold. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M If one drops the positive definiteness requirement of inner product spaces, then one obtains a pseudo-Riemannian metric tensor, which integrates to a pseudo-semimetric. In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. These are used in the geometric study of the theory of relativity, where the tensor is also called the "invariant distance". This page is about the scientific concept of relativity for philosophical or sociological theories about relativity see Relativism.

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