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In mathematics a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric. When a topology can be described by a metric we call the space metrisable. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
Partial plot of a function f. ...
Distance is a numerical description of how far apart things lie. ...
In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...
A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ...
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). In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...
In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ...
Definition A metric on a set X is a function (called the distance function or simply distance) Partial plot of a function f. ...
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: In mathematics, the real numbers may be described informally in several different ways. ...
- d(x, y) ≥ 0 (non-negativity)
- d(x, y) = 0 if and only if x = y (identity of indiscernibles. 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).
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). A negative number is a number that is less than zero, such as −3. ...
The identity of indiscernibles is an ontological principle that states that if there is no way of telling two entities apart then they are one and the same entity. ...
In mathematics, a positive-definite function of a real variable x is a function f:R → C such that for any real numbers x1, ...,xn the n×n matrix A with entries aij = f(xi − xj) is positive semi-definite. ...
In mathematics, a binary relation R over a set X is symmetric if it holds for all a and b in X that if a is related to b then b is related to a. ...
A function f(x) is subadditive if for all x and y in the domain of f. ...
In mathematics, triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides. ...
If two objects are at a distance one mile from each other, it should be possible to construct a road of length one mile between them. ...
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, we call d a translation invariant metric if In geometry, a translation slides an object by a vector a: Ta(p) = p + a. ...
- d(x, y)=d(x + a, y + a)
for all x,y and a in X.
If the second requirement (indiscernability) is dropped, the function is called a pseudometric. Dropping the second and third (symmetry) requirements results in the hemimetric. In mathematics, a metric space is a set (or space) where a distance between points is defined. ...
In mathematics, a hemimetric space is a generalization of a metric space, obtained by removing the requirement of discernability and of symmetry. ...
If the third requirement (symmetry) is dropped (keeping 1,2 and 4), then the function is called a quasimetric. In mathematics, a quasimetric space generalizes the idea of a metric space by removing the requirement of symmetry of the metric. ...
If all but the first requirement are dropped (keeping only positivity and a zero self-distance d(x,x)=0), then the function is called a prametric. A prametric that is symmetric and discerning called a semimetric. In mathematics, a prametric space generalizes the concept of a metric space by not requiring the conditions of symmetry, indiscernability and the triangle inequality. ...
In mathematics, a semimetric space generalizes the concept of a metric space by not requiring the condition of satisfying the triangle inequality. ...
If the triangular inequality is strengthened to
- d(x, z) ≤ max( d(x, y), d(y, z) )
the metric is called ultrametric, see below. In mathematics, an ultrametric space is a special kind of metric space. ...
Notes These conditions express intuitive notions about the concept of distance. 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. Distance is a numerical description of how far apart things lie. ...
Euclid(Greek: ), also known as Euclid of Alexandria, was a Greek mathematician who flourished in Alexandria, Egypt, almost certainly during the reign of Ptolemy I (323–283 BC). ...
Euclid 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  - is a metric defining the same topology. (One can replace
 by any summable sequence (an) of strictly positive numbers.) In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ...
The Euclidean distance of two points x = (x1,...,xn) and y = (y1,...,yn) in Euclidean n-space is computed as It is the ordinary distance between the two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. ...
Manhattan versus Euclidean distance: The red, blue, and yellow lines representing the Manhattan distance all have the same length (12), whereas the green line representing the Euclidian distance has length 6×√2 ≈ 8. ...
In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...
In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...
In functional analysis, a seminorm is a function on a vector space with certain properties characteristic of a measure of length. A space with such a seminorm is then known as a seminormed space. ...
In functional analysis, a topological vector space is called locally convex if its topology is defined by a set of convex neighborhoods of 0. ...
In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ...
See: International System of Units, colloquially called the Metric System, and also metrication. ...
A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ...
In mathematics, a series is a sum of a sequence of terms. ...
In common usage positive is sometimes used in affirmation, as a synonym for yes or to express certainty. Look up Positive on Wiktionary, the free dictionary In mathematics, a number is called positive if it is bigger than zero. ...
Equivalence of metrics For a given set X two metrics d1 and d2 are called topological equivalent (uniformly equivalent) if the identity mapping - id: (X,d1) → (X,d2)
is a homeomorphism (uniform isomorphism). In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ...
In the mathematical field of topology a uniform isomorphism or uniform homeomorphism is a special isomorphism between uniform spaces which respects uniform properties. ...
Relation of norms and metrics Given a normed vector space (X,||.||) we can define a metric on X by 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 any real vector space Rn. ...
- 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 In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
- 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 In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...
- ||x||:=d(x,0)
Similarly, a seminorm induces a pseudometric and a homogeneous, translation invariant pseudometric induces a seminorm. In functional analysis, a seminorm is a function on a vector space with certain properties characteristic of a measure of length. A space with such a seminorm is then known as a seminormed space. ...
Related concepts and alternative axiom systems Some authors use the extended real number line and allow the distance function d to attain the value ∞. Such a metric is called an extended metric. Every extended metric can be rescaled to a finite metric (using d'(x, y) = d(x, y) / (1 + d(x, y)) or d''(x, y) = min(1, d(x, y))) and the two concepts of metric space are therefore equivalent as far as notions of topology (such as continuity or convergence) are concerned. The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ...
A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ...
In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ...
In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ...
A metric is called an ultrametric if it satisfies the following stronger version of the triangle inequality: In mathematics, an ultrametric space is a special kind of metric space. ...
- For all x, y, z in M, d(x, z) ≤ max(d(x, y), d(y, z))
If one drops property 2, one obtains pseudometric spaces. Dropping property 3 instead, one obtains quasimetric spaces. However, losing symmetry in this case, one usually changes property 2 such that both d(x,y)=0 and d(y,x)=0 are needed for x and y to be identified. Dropping property 4 one obtains semimetric spaces. All combinations of the above are possible and are referred to by their according names (such as quasi-pseudo-ultrametric). In mathematics, a metric space is a set (or space) where a distance between points is defined. ...
In mathematics, a quasimetric space (M,d ) is a set M together with a function d : M × M -> R (called a quasimetric) which satisfies the following conditions: d(x, y) ≥ 0 (non-negativity) d(x, y) = 0 if and only if x = y (identity of indiscernibles) d(x, z...
In mathematics, a semimetric space generalizes the concept of a metric space by not requiring the condition of satisfying the triangle inequality. ...
From the 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. 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 maintain these good categorical properties. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...
In topology, approach spaces are a generalization of metric spaces, based on point-to-set distances, instead of point-to-point distances. ...
The requirement that the metric takes values in [0,∞) can also be relaxed to consider metrics with values in other directed sets. 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 mathematics, a directed set is a set A together with a binary relation ≤ having the following properties: a ≤ a for all a in A (reflexivity) if a ≤ b and b ≤ c, then a ≤ c (transitivity) for any two a and b in A, there...
In the mathematical field of topology, a uniform space is a set with a uniform structure. ...
In differential geometry, one considers metric tensors, which can be thought of as "infinitesimal" metric functions, and are defined as inner products on the tangent space with an appropriate differentiability requirement. While these are not metric functions as defined in this article, they induce metric functions by integration. A manifold with a metric tensor is called a Riemannian manifold. 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. These are used in the geometric study of the theory of relativity, where the tensor is also called the "invariant distance". In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...
In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ...
In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...
The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ...
In mathematics, the derivative of a function is one of the two central concepts of calculus. ...
In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ...
On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...
In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ...
In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ...
Two-dimensional analogy of space-time distortion described in General Relativity. ...
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