Uniformly Cauchy sequence
In mathematics, a sequence of functions from a set S to a metric space M is said to be uniformly Cauchy if:
- For all , there exists such that for all : whenever .
Another way of saying this is that as , where the uniform distance between two functions is defined by
Convergence criteria
A sequence of functions {fn} from S to M is pointwise Cauchy if, for each x ∈ S, the sequence {fn(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 fn : S → M tends uniformly to a unique continuous function f : S → M.
Generalization to uniform spaces
A sequence of functions from a set S to a metric space U is said to be uniformly Cauchy if:
- For all and for any entourage , there exists such that whenever .