Editing Metric space (section)

== Metric spaces with additional structure ==
=== Normed vector spaces ===
{{anchor|Norm induced metric|Relation of norms and metrics}}
{{Main|Normed vector space}}
A [[normed vector space]] is a vector space equipped with a ''[[norm (mathematics)|norm]]'', which is a function that measures the length of vectors.  The norm of a vector {{mvar|v}} is typically denoted by <math>\lVert v \rVert</math>.  Any normed vector space can be equipped with a metric in which the distance between two vectors {{mvar|x}} and {{mvar|y}} is given by
<math display="block">d(x,y):=\lVert x-y \rVert.</math>
The metric {{mvar|d}} is said to be ''induced'' by the norm <math>\lVert{\cdot}\rVert</math>.  Conversely,{{sfn|Narici|Beckenstein|2011|pp=47–66}} if a metric {{mvar|d}} on a [[vector space]] {{mvar|X}} is
* translation invariant: <math>d(x,y) = d(x+a,y+a)</math> for every {{mvar|x}}, {{mvar|y}}, and {{mvar|a}} in {{mvar|X}}; and
* [[Absolute homogeneity|{{visible anchor|absolutely homogeneous|Homogeneous metric}}]]: <math>d(\alpha x, \alpha y) = |\alpha| d(x,y)</math> for every {{mvar|x}} and {{mvar|y}} in {{mvar|X}} and real number {{math|α}};
then it is the metric induced by the norm
<math display="block">\lVert x \rVert := d(x,0).</math>
A similar relationship holds between [[seminorm]]s and [[pseudometric space|pseudometric]]s.

Among examples of metrics induced by a norm are the metrics {{math|''d''<sub>1</sub>}}, {{math|''d''<sub>2</sub>}}, and {{math|''d''<sub>∞</sub>}} on <math>\R^2</math>, which are induced by the [[Manhattan norm]], the [[Euclidean norm]], and the [[maximum norm]], respectively.  More generally, the [[Kuratowski embedding]] allows one to see any metric space as a subspace of a normed vector space.

Infinite-dimensional normed vector spaces, particularly spaces of functions, are studied in [[functional analysis]].  Completeness is particularly important in this context: a complete normed vector space is known as a [[Banach space]].  An unusual property of normed vector spaces is that [[linear transformation]]s between them are continuous if and only if they are Lipschitz.  Such transformations are known as [[bounded operator]]s.

=== Length spaces ===
[[File:approximate arc length.svg|thumb|One possible approximation for the arc length of a curve.  The approximation is never longer than the arc length, justifying the definition of arc length as a [[supremum]].]]
{{Main|Intrinsic metric}}
A [[curve]] in a metric space {{math|(''M'', ''d'')}} is a continuous function <math>\gamma:[0,T] \to M</math>.  The [[arc length|length]] of {{math|γ}} is measured by
<math display="block">L(\gamma)=\sup_{0=x_0<x_1<\cdots<x_n=T} \left\{\sum_{k=1}^n d(\gamma(x_{k-1}),\gamma(x_k))\right\}.</math>
In general, this supremum may be infinite; a curve of finite length is called ''rectifiable''.{{sfn|Burago|Burago|Ivanov|2001|loc=Definition 2.3.1}}  Suppose that the length of the curve {{math|γ}} is equal to the distance between its endpoints—that is, it is the shortest possible path between its endpoints.  After reparametrization by arc length, {{math|γ}} becomes a ''[[geodesic]]'': a curve which is a distance-preserving function.{{sfn|Margalit|Thomas|2017}}  A geodesic is a shortest possible path between any two of its points.{{efn|This differs from usage in [[Riemannian geometry]], where geodesics are only locally shortest paths.  Some authors define geodesics in metric spaces in the same way.{{sfn|Burago|Burago|Ivanov|2001|loc=Definition 2.5.27}}{{sfn|Gromov|2007|loc=Definition 1.9}}}}

A ''geodesic metric space'' is a metric space which admits a geodesic between any two of its points.  The spaces <math>(\R^2,d_1)</math> and <math>(\R^2,d_2)</math> are both geodesic metric spaces.  In <math>(\R^2,d_2)</math>, geodesics are unique, but in <math>(\R^2,d_1)</math>, there are often infinitely many geodesics between two points, as shown in the figure at the top of the article.

The space {{mvar|M}} is a ''[[length space]]'' (or the metric {{mvar|d}} is ''intrinsic'') if the distance between any two points {{mvar|x}} and {{mvar|y}} is the infimum of lengths of paths between them.  Unlike in a geodesic metric space, the infimum does not have to be attained.  An example of a length space which is not geodesic is the Euclidean plane minus the origin: the points {{math|(1, 0)}} and {{math|(-1, 0)}} can be joined by paths of length arbitrarily close to 2, but not by a path of length 2.  An example of a metric space which is not a length space is given by the straight-line metric on the sphere: the straight line between two points through the center of the Earth is shorter than any path along the surface.

Given any metric space {{math|(''M'', ''d'')}}, one can define a new, intrinsic distance function {{math|''d''<sub>intrinsic</sub>}} on {{mvar|M}} by setting the distance between points {{mvar|x}} and {{mvar|y}} to be the infimum of the {{mvar|d}}-lengths of paths between them.  For instance, if {{mvar|d}} is the straight-line distance on the sphere, then {{math|''d''<sub>intrinsic</sub>}} is the great-circle distance.  However, in some cases {{math|''d''<sub>intrinsic</sub>}} may have infinite values.  For example, if {{mvar|M}} is the [[Koch snowflake]] with the subspace metric {{mvar|d}} induced from <math>\R^2</math>, then the resulting intrinsic distance is infinite for any pair of distinct points.

=== Riemannian manifolds ===
{{Main|Riemannian manifold}}
A [[Riemannian manifold]] is a space equipped with a Riemannian [[metric tensor]], which determines lengths of [[tangent space|tangent vectors]] at every point.  This can be thought of defining a notion of distance infinitesimally.  In particular, a differentiable path <math>\gamma:[0, T] \to M</math> in a Riemannian manifold {{mvar|M}} has length defined as the integral of the length of the tangent vector to the path:
<math display="block">L(\gamma)=\int_0^T |\dot\gamma(t)|dt.</math>
On a connected Riemannian manifold, one then defines the distance between two points as the infimum of lengths of smooth paths between them.  This construction generalizes to other kinds of infinitesimal metrics on manifolds, such as [[sub-Riemannian manifold|sub-Riemannian]] and [[Finsler manifold|Finsler metrics]].

The Riemannian metric is uniquely determined by the distance function; this means that in principle, all information about a Riemannian manifold can be recovered from its distance function.  One direction in metric geometry is finding purely metric ([[synthetic geometry|"synthetic"]]) formulations of properties of Riemannian manifolds.  For example, a Riemannian manifold is a [[CAT(k) space|{{math|CAT(''k'')}} space]] (a synthetic condition which depends purely on the metric) if and only if its [[sectional curvature]] is bounded above by {{mvar|k}}.{{sfn|Burago|Burago|Ivanov|2001|p=127}}  Thus {{math|CAT(''k'')}} spaces generalize upper curvature bounds to general metric spaces.

=== Metric measure spaces ===

Real analysis makes use of both the metric on <math>\R^n</math> and the [[Lebesgue measure]].  Therefore, generalizations of many ideas from analysis naturally reside in [[metric measure space]]s: spaces that have both a [[measure (mathematics)|measure]] and a metric which are compatible with each other.  Formally, a ''metric measure space'' is a metric space equipped with a [[Borel regular measure]] such that every ball has positive measure.{{sfn|Heinonen|2007|p=191}}  For example Euclidean spaces of dimension {{mvar|n}}, and more generally {{mvar|n}}-dimensional Riemannian manifolds, naturally have the structure of a metric measure space, equipped with the [[Lebesgue measure]].  Certain [[fractal]] metric spaces such as the [[Sierpiński gasket]] can be equipped with the α-dimensional [[Hausdorff measure]] where α is the [[Hausdorff dimension]].  In general, however, a metric space may not have an "obvious" choice of measure.

One application of metric measure spaces is generalizing the notion of [[Ricci curvature]] beyond Riemannian manifolds.  Just as {{math|CAT(''k'')}} and [[Alexandrov space]]s generalize sectional curvature bounds, [[RCD space]]s are a class of metric measure spaces which generalize lower bounds on Ricci curvature.<ref>{{cite journal |last1=Gigli |first1=Nicola |title=Lecture notes on differential calculus on RCD spaces |journal=Publications of the Research Institute for Mathematical Sciences |date=18 October 2018 |volume=54 |issue=4 |pages=855–918 |doi=10.4171/PRIMS/54-4-4 |arxiv=1703.06829|s2cid=119129867 }}</ref>