Metric Space - Examples of Metric Spaces

Examples of Metric Spaces

• Ignoring mathematical details, for any system of roads and terrains the distance between two locations can be defined as the length of the shortest route connecting those locations. To be a metric there shouldn't be any one-way roads. The triangle inequality expresses the fact that detours aren't shortcuts. Many of the examples below can be seen as concrete versions of this general idea.
• The real numbers with the distance function given by the absolute difference, and more generally Euclidean -space with the Euclidean distance, are complete metric spaces. The rational numbers with the same distance also form a metric space, but are not complete.
• The positive real numbers with distance function is a complete metric space.
• Any normed vector space is a metric space by defining, see also metrics on vector spaces. (If such a space is complete, we call it a Banach space.) Examples:
  • The Manhattan norm gives rise to the Manhattan distance, where the distance between any two points, or vectors, is the sum of the differences between corresponding coordinates.
  • The maximum norm gives rise to the Chebyshev distance or chessboard distance, the minimal number of moves a chess king would take to travel from to .
• The British Rail metric (also called the Post Office metric or the SNCF metric) on a normed vector space is given by for distinct points and, and . More generally can be replaced with a function taking an arbitrary set to non-negative reals and taking the value at most once: then the metric is defined on by for distinct points and, and . The name alludes to the tendency of railway journeys (or letters) to proceed via London (or Paris) irrespective of their final destination.
• If is a metric space and is a subset of, then becomes a metric space by restricting the domain of to .
• The discrete metric, where if and otherwise, is a simple but important example, and can be applied to all non-empty sets. This, in particular, shows that for any non-empty set, there is always a metric space associated to it. Using this metric, any point is an open ball, and therefore every subset is open and the space has the discrete topology.
• A finite metric space is a metric space having a finite number of points. Not every finite metric space can be isometrically embedded in a Euclidean space.
• The hyperbolic plane is a metric space. More generally:
• If is any connected Riemannian manifold, then we can turn into a metric space by defining the distance of two points as the infimum of the lengths of the paths (continuously differentiable curves) connecting them.
• If is some set and is a metric space, then, the set of all bounded functions (i.e. those functions whose image is a bounded subset of ) can be turned into a metric space by defining for any two bounded functions and (where is supremum. This metric is called the uniform metric or supremum metric, and If is complete, then this function space is complete as well. If X is also a topological space, then the set of all bounded continuous functions from to (endowed with the uniform metric), will also be a complete metric if M is.
• If is an undirected connected graph, then the set of vertices of can be turned into a metric space by defining to be the length of the shortest path connecting the vertices and . In geometric group theory this is applied to the Cayley graph of a group, yielding the word metric.
• The Levenshtein distance is a measure of the dissimilarity between two strings and, defined as the minimal number of character deletions, insertions, or substitutions required to transform into . This can be thought of as a special case of the shortest path metric in a graph and is one example of an edit distance.
• Given a metric space and an increasing concave function such that if and only if, then is also a metric on .
• Given an injective function from any set to a metric space, defines a metric on .
• Using T-theory, the tight span of a metric space is also a metric space. The tight span is useful in several types of analysis.
• The set of all by matrices over some field is a metric space with respect to the rank distance .
• The Helly metric is used in game theory.