Uniformly Cauchy sequence

Uniformly Cauchy sequence

In mathematics, a sequence of functions {f_{n}} from a set "S" to a metric space "M" is said to be uniformly Cauchy if:

* For all xin S and for all varepsilon > 0, there exists N>0 such that d(f_{n}(x), f_{m}(x)) < varepsilon whenever m, n > N.

Another way of saying this is that d_u (f_{n}, f_{m}) o 0 as m, n o infty, where the uniform distance d_u between two functions is defined by

:d_{u} (f, g) := sup_{x in S} d (f(x), g(x)).

Convergence criteria

A sequence of functions {"f"n} from "S" to "M" is pointwise Cauchy if, for each "x" &isin; "S", the sequence {"f"n("x")} is a Cauchy sequence in "M". This is a weaker condition than being uniformly Cauchy. Nevertheless, if the metric space "M" is complete, then any pointwise Cauchy sequence converges pointwise to a function from "S" to "M". Similarly, any uniformly Cauchy sequence will tend uniformly to such a function.

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:

* Let "S" be a topological space and "M" a complete metric space. Then any uniformly Cauchy sequence of continuous functions "f"n : "S" &rarr; "M" tends uniformly to a unique continuous function "f" : "S" &rarr; "M".

Generalization to uniform spaces

A sequence of functions {f_{n}} from a set "S" to a metric space "U" is said to be uniformly Cauchy if:

* For all xin S and for any entourage varepsilon, there exists N>0 such that (f_{n}(x), f_{m}(x)) in varepsilon whenever m, n > N.

ee also

*Modes of convergence (annotated index)


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