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On a compact metric space, equicontinuity plus pointwise boundedness implies uniform boundedness.
Equicontinuity–boundedness criterion: Let be a compact metric space and let be a family of functions . Assume is an equicontinuous family and is pointwise bounded on . Then is uniformly bounded on .
This lemma is a standard compactness-based tool used alongside Arzelà–Ascoli when working in the supremum norm.
A metric space is a pair where is a set and is a metric on .
Every metric space determines a topological space via the metric-induced topology, whose basic neighborhoods are open balls.
A family of functions from a metric space to a metric space is an equicontinuous family (or is equicontinuous on ) if it is equicontinuous at each point: for every and every there exists such that for all and all ,
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.