# 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.
References in periodicals archive ?
By applying the Arzela-Ascoli theorem, we can guarantee that T(D) is relatively compact, which means T is compact.
Zimmer, An asymmetric Arzela-Ascoli theorem, Topology and it's Applications, 154(2007), 2312-2322.
n], by Arzela-Ascoli theorem there exist the subsequences (still denoted) [u.
The Arzela-Ascoli theorem then implies that T is completely continuous.
As a consequence of Step 1-3, together with Arzela-Ascoli theorem, we can conclude that p: [C.
According to the Arzela-Ascoli theorem, it is not difficult to show that [K.
The functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are uniformly bounded and equicontinuous; due to the Arzela-Ascoli theorem a uniformly convergent subsequence can be selected (but may be not unique).
The Arzela-Ascoli theorem guarantees the existence of the subsequence S of [N.
T]; then by applying the Arzela-Ascoli theorem on time scales [1], we know that F[bar.
lambda]] is even completely continuous for fixed [lambda] by Arzela-Ascoli theorem.
As a consequence of steps 1 to 3 together with the Arzela-Ascoli theorem, we can conclude that N : D [right arrow] D is continuous and compact.
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