Continuous mappings | Mathematical analysis

Equicontinuity

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 metric space or locally compact space is continuous. If, in addition, fn are holomorphic, then the limit is also holomorphic. The uniform boundedness principle states that a pointwise bounded family of continuous linear operators between Banach spaces is equicontinuous. (Wikipedia).

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Uniform convergence | Topological space | Metric space | Convergence of random variables | Countable set | Uniform boundedness principle | Operator norm | Heine–Borel theorem | Convex hull | Separable space | Continuous function | Mathematical analysis | Topological group | Topological vector space | Banach space | Barrelled space | Mean value theorem | Locally compact space | Filter (set theory) | Absolutely convex set | Seminorm | Uniform space | Neighbourhood (mathematics) | Sign function | Uniform continuity | Continuous functions on a compact Hausdorff space | Polar set | Uniform norm | Balanced set | Analytic function | Arzelà–Ascoli theorem | Complete metric space