equicontinuity
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In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus sequences of functions.
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Article
16 sectionsContents
- Equicontinuity between metric spaces
- Examples
- Counterexamples
- Equicontinuity of maps valued in topological groups
- Equicontinuous linear maps
- Characterization of equicontinuous linear maps
- Characterization of equicontinuous linear functionals
- Properties of equicontinuous linear maps
- Properties of equicontinuous linear functionals
- Equicontinuity and uniform convergence
- Generalizations
- Equicontinuity in topological spaces
- Stochastic equicontinuity
- See also
- Notes
- References
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus sequences of functions.
Equicontinuity appears in the formulation of Ascoli's theorem, which states that a subset of C(X), the space of continuous functions on a compact Hausdorff space X, is compact if and only if it is closed, pointwise bounded and equicontinuous. As a corollary, a sequence in C(X) is uniformly convergent if and only if it is equicontinuous and converges pointwise to a function (not necessarily continuous a-priori). In particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions fn on either a metric space or a locally compact space is continuous. If, in addition, fn are holomorphic, then the limit is also holomorphic.