FIVE MAJOR RESULTS IN ANALYSIS AND TOPOLOGY Aaron ...

CHAPTER 2
The Ascoli-Arzelà Theorem
One often finds that significant mathematical results which establish the existence of
some object in a space often rest on crucial properties of the space in question, such as
compactness. Naturally, any theorem which establishes a set of necessary and sufficient
conditions for compactness of a space could spawn any number of existential results. The
Ascoli-Arzelà theorem is one such example. It claims that a subset D in the space of
continuous functions from one compact metric space to another is compact if, and only if,
its members are equicontinuous.
The proof of the Ascoli-Arzelà theorem will be abbreviated by the following result.
Theorem (Heine-Cantor). 1 Let (X,ρX) be a compact metric space, and (Y,ρY ) a metric
space. Then every f ∈ C(X,Y ) is uniformly continuous.
Proof. Suppose, on the contrary, that there existed an f0 ∈ C(X,Y ) and an ǫ0 > 0 such
that for every δ > 0, one can find x,y ∈ X so that ρX(x,y) < δ, but ρY (f0(x),f0(y)) ≥ ǫ0.
For n ∈ N, choose xn,yn so that ρX(xn,yn) < 1
n , but ρY (f0(xn),f0(yn)) ≥ ǫ0. Since
X is compact, one can find a convergent subsequence {xnk } of {xn}. Let x = lim xnk
k→∞ .
Similarly, choose a convergent subsequence {ynk } of {yn}, and set y = lim ynk . Since the
k→∞
sequence xn1,yn1,xn2,yn2,... is fundamental in a compact space, let g be its limit. Since every
subsequence of a convergent sequence converges to the same limit, we have x = lim
k→∞ xnk =
g = lim
k→∞ ynk
= y. Then ρX(xnk ,ynk ) < 1
nk and ρY (f0(xnk ),f0(ynk )) ≥ ǫ0 for every k ∈ N.
This contradicts our assumption that f0 ∈ C(X,Y ).
1 The proof of this theorem is similar to that found in [12]
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