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On a compact metric space, an equicontinuous, pointwise bounded family of real-valued functions is uniformly bounded.
Let be a compact metric space and let . If is equicontinuous and pointwise bounded, meaning
then is uniformly bounded. Thus there is such that
This lemma is a standard step in the proof of the Arzelà–Ascoli theorem.
A compact set is a subset of a topological space such that every open cover of contains a finite subcover: whenever is a family of open sets with , there exist indices such that
A family of functions from a metric space to a metric space is equicontinuous on if, for every and , there exists such that, for every and ,
Equicontinuity provides uniform control of continuity across the family and is a key hypothesis (together with pointwise boundedness) in the Arzelà–Ascoli theorem for subsets of spaces of continuous functions equipped with the uniform metric.
A family of functions is pointwise bounded if for every the set of values is bounded in , equivalently
A family of functions is uniformly bounded if there exists such that
Equivalently, if every is bounded, then in terms of the supremum norm.
Arzelà–Ascoli theorem: Let be a compact metric space, and let be continuous for all . If the sequence is uniformly bounded and equicontinuous, then there exists a subsequence and a continuous function such that uniformly on .
In the language of the space of continuous functions with the supremum norm, this is a compactness criterion for extracting uniformly convergent subsequences from equicontinuous, uniformly bounded families.