# Ascoli's theorem

(redirected from Arzela-Ascoli 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 ?
From a consequence of Steps 1 to 3, together with the Arzela-Ascoli theorem, we deduce that the operator A : C [right arrow] C is completely continuous.
In consequence, it follows by the Arzela-Ascoli theorem that the operator [[PSI].sub.k] is completely continuous.
Without loss of generality, we may suppose that, by Arzela-Ascoli theorem and [mathematical expression not reproducible].
By means of the Arzela-Ascoli theorem, T : P [right arrow] P is completely continuous.
The Arzela-Ascoli theorem implies that [mathematical expression not reproducible] is relatively compact.
As a consequence of Steps 1 to 3 and asumption (iv) of Theorem 3.1 together with the Arzela-Ascoli theorem, we can deduce that [PHI] : PC(0, b; X) [right arrow] PC(0, b; X) is a completely continuous operator.
Hence, by Arzela-Ascoli Theorem [??] is compact and by virtue of Proposition 9, [??] is Y-Lipschitz with zero constant.
Thus the sequence {[K.sub.P,Q]N[x.sub.n](t)} is equicontinuous on [0, 1] and by Arzela-Ascoli Theorem is convergent.
Thus, by the Arzela-Ascoli theorem there is a subsequence of [(H([N.sub.f]([u.sub.n]))).sub.n], which we call [mathematical expression not reproducible], which is convergent in C.
Then by Arzela-Ascoli Theorem, {[f.sub.k]} has a subsequence {fk.}, say it {fk} again, such that each {[f.sup.(r).sub.k]}, 0 [less than or equal to] r [less than or equal to] n - 1 is uniformly convergent and hence is uniformly Cauchy on X.
As a consequence of steps 1 to 4 together with the Arzela-Ascoli theorem, we can conclude that N is continuous and compact.
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