# Ascoli's theorem

## Ascoli's theorem

[as′kō‚lēz ‚thir·əm]
(mathematics)
The theorem that a set of uniformly bounded, equicontinuous, real-valued functions on a closed set of a real Euclidean n-dimensional space contains a sequence of functions which converges uniformly on compact subsets.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
References in periodicals archive ?
By Ascoli's theorem there exists a uniformly convergent subsequence [w.sub.i] (t + [t.sub.k]) [subset or equal to] [w.sub.i] (t + [t'.sub.k])(i = 1, 2, 3, 4) such that, for any e > 0, there exists a K(e) > 0 with the property that if m, k [greater than or equal to] K([epsilon]), then
Therefore by Ascoli's theorem, the family {[[integral].sub.D] [G.sub.D] (x, y) f (y, v(y)) dy, v [member of] [LAMBDA] becomes relatively compact in [C.sub.0](D).
By Ascoli's theorem, there exists a uniformly convergent subsequence {[u.sub.i](t + [t.sub.k])} [subset or equal to] {[u.sub.i](t + [t'.sub.k])} such that for any [epsilon] > 0, there exists a K([epsilon]) > 0 with the property that if m, k [greater than or equal to] K([epsilon]), then
He covers real numbers and limits, including the concepts of infinity and sequences, topology, including the Cantor set and fractals, and then progresses to calculus, including the Riemann integral, sequences of functions, power and Fourier series and the exponential function, closing with metric spaces, including Ascoli's theorem.
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