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| In [[mathematics]], a [[sequence]] of [[function (mathematics)|function]]s <math>\{f_{n}\}</math> from a set ''S'' to a metric space ''M'' is said to be '''uniformly Cauchy''' if:
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| * For all <math>\varepsilon > 0</math>, there exists <math>N>0</math> such that for all <math>x\in S</math>: <math>d(f_{n}(x), f_{m}(x)) < \varepsilon</math> whenever <math>m, n > N</math>.
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| Another way of saying this is that <math>d_u (f_{n}, f_{m}) \to 0</math> as <math>m, n \to \infty</math>, where the uniform distance <math>d_u</math> between two functions is defined by
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| :<math>d_{u} (f, g) := \sup_{x \in S} d (f(x), g(x)).</math>
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| == Convergence criteria ==
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| A sequence of functions {''f''<sub>n</sub>} from ''S'' to ''M'' is '''pointwise''' Cauchy if, for each ''x'' ∈ ''S'', the sequence {''f''<sub>n</sub>(''x'')} is a [[Cauchy sequence]] in ''M''. This is a weaker condition than being uniformly Cauchy. Nevertheless, if the metric space ''M'' is [[complete metric space|complete]], then any pointwise Cauchy sequence converges pointwise to a function from ''S'' to ''M''. Similarly, any uniformly Cauchy sequence will tend [[uniform convergence|uniformly]] to such a function.
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| The uniform Cauchy property is frequently used when the ''S'' is not just a set, but a [[topological space]], and ''M'' is a complete metric space. The following theorem holds:
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| * Let ''S'' be a topological space and ''M'' a complete metric space. Then any uniformly Cauchy sequence of [[continuous function]]s ''f''<sub>n</sub> : ''S'' → ''M'' tends [[uniform convergence|uniformly]] to a unique continuous function ''f'' : ''S'' → ''M''.
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| == Generalization to uniform spaces ==
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| A [[sequence]] of [[function (mathematics)|function]]s <math>\{f_{n}\}</math> from a set ''S'' to a metric space ''U'' is said to be '''uniformly Cauchy''' if:
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| * For all <math>x\in S</math> and for any [[Uniform space|entourage]] <math>\varepsilon</math>, there exists <math>N>0</math> such that <math>d(f_{n}(x), f_{m}(x)) < \varepsilon</math> whenever <math>m, n > N</math>.
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| ==See also==
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| *[[Modes of convergence (annotated index)]]
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| [[Category:Functional analysis]]
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| [[Category:Convergence (mathematics)]]
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| {{mathanalysis-stub}}
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