Metric space

In mathematics, a metric space is a set on which a metric is defined. A metric (also called a distance function ) is a function that assigns a non-negative real value to two elements of the space , which can be understood as the distance between the two elements.

Formal definition

Be any set. A figure called in metric , if for any element , and by the following axioms are satisfied: ${\displaystyle X}$ ${\displaystyle d\colon X\times X\to \mathbb {R} }$ ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$ ${\displaystyle X}$

(1) Positive definiteness:	${\displaystyle d\left(x,y\right)\geq 0}$     and     , ${\displaystyle d\left(x,y\right)=0\Leftrightarrow x=y}$
(2) symmetry :	${\displaystyle d\left(x,y\right)=d(y,x)}$,
(3) triangle inequality :	${\displaystyle d\left(x,y\right)\leq d(x,z)+d(z,y)}$.

The requirement can be omitted because it follows from the others, there ${\displaystyle d(x,y)\geq 0}$

${\displaystyle 0={\frac {1}{2}}d(x,x)\leq {\frac {1}{2}}(d(x,y)+d(y,x))={\frac {1}{2}}(d(x,y)+d(x,y))=d(x,y).}$

Basic concepts

${\displaystyle (X,d)}$is called metric space if a metric is open . Some authors also demand that there should be a non-empty set . In practice, it is mostly referred to as the metric space alone if it is clear from the context that the metric is used in this space . ${\displaystyle d}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle d}$

An isometry is a mapping that maps two metric spaces to one another and thereby preserves the metric - i.e. the distances between two points.

Generated topology

The open spheres in a metric space create (as a basis ) a topology , the topology induced by the metric.

Generalizations and specializations

By weakening, omitting or tightening one or more of the conditions (1) to (3), various generalizations or specializations result. The names for the generalizations are unfortunately not standardized for all areas of mathematics in which they are used. Specifically, a semimetric in functional analysis is understood to be something different than in topology (see below).

Ultrametric

→ Main article : Ultrametrics

If the condition of the triangle inequality is tightened to the effect that the distance must not be longer than the longer of the two distances and (with any ), the term ultrametric is obtained . ${\displaystyle d(x,y)}$ ${\displaystyle d(x,z)}$ ${\displaystyle d(z,y)}$ ${\displaystyle z}$

Pseudometrics

If the condition is omitted, the term pseudometrics is obtained . In functional analysis, the term half-metric or semimetric is also used for this. In pseudometric spaces , non-identical points can have a distance of 0. A pseudometric is positive semidefinite; H. Distances are always greater than or equal to 0. ${\displaystyle d\left(x,y\right)=0\Rightarrow x=y}$

Quasi-metric

If the symmetry is dispensed with, the term quasi-metric is obtained. A metric can be used to generate from a quasi- metric . ${\displaystyle d'}$ ${\displaystyle d(x,y):={\tfrac {1}{2}}(d'(x,y)+d'(y,x))}$ ${\displaystyle X}$

Non-Archimedean metrics

If the triangle inequality is weakened or tightened, then non-Archimedean metrics are obtained. An example is for one or the ultrametric. ${\displaystyle d(x,y)\leq K(d(x,z)+d(z,y))}$ ${\displaystyle K>1}$

In topology, metrics with no triangle inequality are sometimes referred to as semimetrics.

Premetric

If only non-negativity and condition (1) are required, then one speaks of a premetric. To be carried, for example, ${\displaystyle \mathbb {R} }$

${\displaystyle d(x,y)={\begin{cases}1&{\text{für }}x>y\\|x-y|&{\text{sonst }}\end{cases}}}$

such a premetric is defined.

Examples

Metrics generated by standards

Every norm on a vector space induced by the definition

${\displaystyle d(x,y)\equiv \|x-y\|}$

a metric. Thus every normalized vector space (and especially every interior product space , Banach space or Hilbert space ) is a metric space.

A metric that is derived from a p norm is also called a Minkowski metric . Important special cases are

• the Manhattan metric too ${\displaystyle p=1}$
• the Euclidean metric too ${\displaystyle p=2}$
• the maximum metric too ${\displaystyle p=\infty }$

Further examples of norms (and thus also of metrics) can be found in the article Norm (mathematics) .

For example, the metrics of the following important spaces are derived from a p -norm:

• the one-dimensional space of real or complex numbers with the absolute amount as norm (with any ) and the amount metric given by it ${\displaystyle p}$

${\displaystyle d(x,y)=|x-y|}$

• the Euclidean space with its by the Pythagorean theorem given Euclidean metric (for Euclidean norm for ) ${\displaystyle p=2}$

${\displaystyle d(x,y)={\sqrt {(x_{1}-y_{1})^{2}+\dotsb +(x_{n}-y_{n})^{2}}}}$

Occasionally, a metric is called a Fréchet metric

${\displaystyle d(x,y)=\rho (x-y)}$

which is induced by a function which has most of the properties of a standard but is not homogeneous. ${\displaystyle \rho }$

Metrics not generated by standards

• A trivial metric, the so-called uniformly discrete metric (which is even an ultrametric), can be defined on each set by

  ${\displaystyle d(x,y)={\begin{cases}0&{\text{für }}x=y\\1&{\text{für }}x\neq y\end{cases}}}$

It induces the discrete topology .

• On is defined by a metric. Regarding this metric it is not exhaustive. So is z. B. the sequence a - Cauchy sequence that does not converge into. The topology generated by this metric is the same as the standard topology , but the uniform structures induced by the two metrics are obviously different. ${\displaystyle \mathbb {R} }$ ${\displaystyle \delta (x,y)=|\arctan(x)-\arctan(y)|\,}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle (n)_{n\in \mathbb {N} }}$ ${\displaystyle \delta }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$
• The Riemannian metric , which turns a differentiable manifold into a Riemannian manifold , is generally not induced by a norm . Examples of this:
  • the natural metric on a spherical surface in which the great circle is the shortest connection ( geodesic ) between two points;
  • the improper metric in the Minkowski space of the special theory of relativity , in which time-like distances are given by [(Δt) ² - (Δx / c) ² - (Δy / c) ² - (Δz / c) ² ] ^1/2 and spatially similar distances [(Δx) ² + (Δy) ² + (Δz) ² - (Δct) ² ] ^{1/2 are} given; ${\displaystyle \mathbb {R} \times \mathbb {R} ^{3}}$
  • the generalization of this metric in general relativity, which depends on the distribution of matter .
• The French railroad metric is a popular practice example of a non-standard induced metric. It is defined with reference to a marked point (“ Paris ”) as follows: The distance between two different points whose connecting line runs through is their distance under the usual Euclidean metric. The distance between two different points whose connecting line does not run through is the sum of their distances from . ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle P}$
• The Hausdorff metric measures the distance between subsets , not elements , of a metric space; it could be called a second-degree metric, because it uses a first-degree metric between the elements of metric space.
• The Hamming distance is a metric in the code space that indicates the difference in character strings (of the same length) .

Classification in the hierarchy of mathematical structures

Hierarchy of topological
spaces and the associated structures

Euclidean space	Has	Scalar product
is		induced
Standardized space	Has	standard
is		induced
Metric space	Has	Metric
is		induced
Uniform room	Has	Uniform structure
is		induced
Topological space	Has	topology

Metrics give a room a global and a local mathematical structure . The global structure is expressed in geometric properties such as the congruence of figures. The local metric structure, i.e. the definition of small distances, enables the introduction of differential operations under certain additional conditions.

The term “ topological space ” generalizes the term “metric space”: Every metric space is a topological space with the topology that is induced by the metric (see environment ). Every metric room is a Hausdorff room .

A topological space is called metrizable if it is homeomorphic to a metric space . A topological space (X, T) can therefore be metrized if a metric d exists on X which induces the topology T.

A full metric space is a metric space in which each Cauchy sequence converges . See the detailed article full space . A completely normalized vector space is called a Banach space . A Banach space whose norm is induced by a scalar product is called a Hilbert space .
In the absence of structural prerequisites, the Cauchy sequence and completeness cannot be defined on general topological spaces. If at least one uniform structure exists , then there is a Cauchy filter and the possibility of completion, which assigns a limit value to each Cauchy filter.

history

Metric spaces were first used in 1906 by Maurice Fréchet in the work Sur quelques points du calcul fonctionnel . The term metric space was coined by Felix Hausdorff .

literature

• Otto Forster : Analysis. Volume 2: differential calculus in R ⁿ . Ordinary differential equations. 7th improved edition. Vieweg , Wiesbaden 2006, ISBN 3-8348-0250-6 ( Vieweg study. Basic course in mathematics ).
• Athanase Papadopoulos: Metric Spaces, Convexity and Nonpositive Curvature. European Mathematical Society , Zurich 2004, ISBN 3-03719-010-8 .
• Boto von Querenburg : Set theoretical topology . 3., rework. and exp. Edition. Springer , Berlin / Heidelberg / New York 2001, ISBN 978-3-540-67790-1 , doi : 10.1007 / 978-3-642-56860-2 .

Individual evidence

1. ^ Franz Lemmermeyer : Topology . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 978-3-8274-0439-8 .