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Uniformly Cauchy sequence

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

Contents

  • For all ε > 0 , there exists N > 0 such that for all x ∈ S : d ( f n ( x ) , f m ( x ) ) < ε whenever m , n > N .

    • Another way of saying this is that d u ( f n , f m ) → 0 as m , n → ∞ , where the uniform distance d u between two functions is defined by

    d u ( f , g ) := sup x ∈ S d ( f ( x ) , g ( x ) ) .

    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.

    In general a sequence can be pointwise Cauchy and not pointwise convergent, or it can be uniformly Cauchy and not uniformly convergent. 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 { f n } from a set S to a metric space U is said to be uniformly Cauchy if:

  • For all x ∈ S and for any entourage ε , there exists N > 0 such that d ( f n ( x ) , f m ( x ) ) < ε whenever m , n > N .

    • References

    Uniformly Cauchy sequence Wikipedia
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