The Arzela-Ascoli Theorem is a primary result of mathematical analysis providing necessary and sufficient conditions to determine whether every sequence of a given set of real-valued continuous functions illustrated on a closed and bounded interval comprises a uniformly convergent subsequence. Here, the primary condition is the equicontinuity of the group of functions. Let’s have a look at the statement and proof of the Arzela-Ascoli theorem given in this article.

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Arzela-Ascoli Theorem in Metric Space

If a sequence {f_n}₁^∞ in C(X) is bounded and equicontinuous, then it holds a uniformly convergent subsequence.

This statement can be written in mathematical terms as follows:

(a) “F ⊂ C(X) is bounded” means there exists a positive constant M < ∞ such that |f(x)| ≤ M for each x ∈ X for all f ∈ F, and

(b) “F ⊂ C(X) is equicontinuous” means for every ε > 0 there exists δ > 0 such that for x, y ∈ X:

d(x, y) < δ ⇒ |f(x) − f(y)| < ε for all f ∈ F

Here, d is the metric on X, and δ depends only on ε.

Arzela-Ascoli Theorem Proof

First, we need to show that the compact metric space X is separable, i.e., has a finite dense subset S.

Given a positive integer n and a point x ∈ X.

Let B(x, 1/n) = {y ∈ X : d(x, y) < 1/n} be the open ball with radius 1/n, centred at x.

For a given value of n, the set of all these balls as x ranges through X is an open cover of x since X is compact.

Thus, there is a countable subcollection that also covers X.

Let S_n be the set of centres of the balls in the finite subcollection defined above.

Thus, S_n is a finite subset of X, consequently “1/n-dense” in that each point of X lies within 1/n of a point of the set Sn.

Therefore, the union S of all the sceneries S_n is finite and dense in X.

Let us find a subsequence of {f_n} that converges pointwise on S.

Let {x₁, x₂,…} be the elements of S.

As we know. then the numerical sequence {f_n(x₁)}_n=1^∞ is bounded.

Thus, by Bolzano-Weierstrass, we can say that it has a convergent subsequence, and this can be written using double subscripts as:

{f₁,n(x₁)}_n=1^∞.

In the same way, we can show that the numerical sequence {f₁,n(x₂)}_n=1^∞ is bounded, so it has a convergent subsequence {f₂,n(x₂)}_n=1^∞.

From the above, we can say that the sequence of functions {f₂,n}_n=1^∞, since it is a subsequence of {f₁,n}_n=1^∞, converges at both x₁ and x₂.

Moving in this technique, we obtain a finite collection of subsequences of the initial sequence, as:

f_1,1 f_1,2 f_1,3 ···

f_2,1 f_2,2 f_2,3 ···

f_3,1 f_3,2 f_3,3 ···

. . . ………

. . . ………

f_n,1 f_n,2 f_n,3 ….

Here, the sequence of the n-th row converges at x₁,…, x_n, and every row is a subsequence of its previous row.

Therefore, the diagonal sequence in the above, i.e., {f_n,n} is a subsequence of the initial sequence {f_n} that converges at each point of S.

Let {g_n} be the diagonal subsequence that is convergent at each point of the dense set S that is created from the overhead step.

Let ε > 0 and δ > 0 then by equicontinuity of the original sequence, so that d(x, y) < δ which implies |g_n(x) − g_n(y)| < ε/3 for each x, y ∈ x and n is a positive integer.

Specify M > 1/δ so that the countable subset SM ⊂ S that we produced in the initial steps is δ-dense in X.

As {g_n} converges at each point of SM, there exists N > 0 such that;

n, m > N ⇒ |g_n(s) − g_m(s)| < ε/3 for all s ∈ SM.

Consider x ∈ X then x lies within δ of some s ∈ SM, so if n, m > M:

|g_n(x) − g_m(x)| ≤ |g_n(x) − g_n(s)| + |g_n(s) − g_m(s)| + |g_m(s) − g_m(x)|

In RHS, the first and last terms are < ε/3 by our preference of δ.

Thus, for a given ε > 0 we can produce N such that for each x ∈ X, m, n > N ⇒ |g_n(x) − g_m(x)| < ε/3 + ε/3 + ε/3 = ε.

Therefore, on X the subsequence {g_n} of {f_n} is uniformly Cauchy, and uniformly convergent.

Hence proved.

Arzela-Ascoli Theorem Applications

The Arzela-Ascoli theorem is the basis of many mathematical proofs, including

• Functional analysis
• Peano’s existence theorem in the theory of ordinary differential equations,
• Montel’s theorem in complex analysis
• Peter–Weyl theorem in harmonic analysis, etc.

Apart from the above mentioned applications, we have many other applications of Arzela-Ascoli in different mathematical areas.