Editing Metric space (section)

=== Distances between mathematical objects ===
In modern mathematics, one often studies spaces whose points are themselves mathematical objects.  A distance function on such a space generally aims to measure the dissimilarity between two objects.  Here are some examples:
* '''Functions to a metric space.''' If {{mvar|X}} is any set and {{mvar|M}} is a metric space, then the set of all [[bounded function]]s <math>f \colon X \to M</math> (i.e. those functions whose image is a [[bounded subset]] of <math>M</math>) can be turned into a metric space by defining the distance between two bounded functions {{mvar|f}} and {{mvar|g}} to be <math display="block">d(f,g) = \sup_{x \in X} d(f(x),g(x)).</math> This metric is called the [[uniform metric]] or supremum metric.{{sfn|Ó Searcóid|2006|p=107}}  If {{mvar|M}} is complete, then this [[function space]] is complete as well; moreover, if {{mvar|X}} is also a topological space, then the subspace consisting of all bounded [[continuous function (topology)|continuous]] functions from {{mvar|X}} to {{mvar|M}} is also complete.  When {{mvar|X}} is a subspace of <math>\R^n</math>, this function space is known as a [[classical Wiener space]].
* '''[[String metric]]s and [[edit distance]]s.''' There are many ways of measuring distances between [[string (computer science)|strings of characters]], which may represent [[sentence (linguistics)|sentence]]s in [[computational linguistics]] or [[Code word (communication)|code word]]s in [[coding theory]].  ''Edit distances'' attempt to measure the number of changes necessary to get from one string to another.  For example, the [[Hamming distance]] measures the minimal number of substitutions needed, while the [[Levenshtein distance]] measures the minimal number of deletions, insertions, and substitutions; both of these can be thought of as distances in an appropriate graph.
* [[Graph edit distance]] is a measure of dissimilarity between two [[Graph (discrete mathematics)|graphs]], defined as the minimal number of [[Graph operations|graph edit operations]] required to transform one graph into another.
* [[Wasserstein metric]]s measure the distance between two [[measure (mathematics)|measure]]s on the same metric space.  The Wasserstein distance between two measures is, roughly speaking, the [[optimal transport|cost of transporting]] one to the other.
* The set of all {{mvar|m}} by {{mvar|n}} [[matrix (mathematics)|matrices]] over some [[field (mathematics)|field]] is a metric space with respect to the [[Rank (linear algebra)|rank]] distance <math>d(A,B) = \mathrm{rank}(B - A)</math>.
* The [[Helly metric]] in [[game theory]] measures the difference between [[strategy (game theory)|strategies]] in a game.