In the textbook Topology by Munkers, Ascoli's theorem (45.4 and 47.1) refers to the fact that certain equicontinuous families of functions lies in a compact set, while Arzela's theorem is about certain equicontinuous sequences having a convergent subsequence.

In contrast, in the convention I'm familiar to, the term the Ascoli-Arzela theorem refers to a single theorem of the same vein. This convention is used in, e.g., the Wikipedia article about this matter.

Question: which kind of result does the term the Ascoli-Arzela theorem usually refer to? Is it about families of functions, or about sequences? Is the convention in Munkers's book where Ascoli's and Arzela's theorems refer to different things common?

(I'm mostly interested in the situations where the domain of the functions in question is more general than closed intervals in $\Bbb R$, which I believe what Ascoli and Arzela actually proved was about.)

  • $\begingroup$ Why do you want a definite answer? The two statements are equivalent. $\endgroup$ – Eric Wofsey Nov 11 '16 at 3:30
  • $\begingroup$ @EricWofsey Because they are, at least prima facie, different statements. (All true theorems are equivalent ;-) ) $\endgroup$ – Pteromys Nov 11 '16 at 3:34

Ascoli-Arzela (or Arzela-Ascoli, as some authors prefer) as I have usually seen it referred to encompasses the following result and its special cases:

Let $X$ be a compact Hausdorff space. Let $C(X)$ be the Banach space of real-valued continuous functions on $X$ under the supremum norm. A family $F\subset C(X)$ of functions is relatively compact (aka precompact, i.e. has compact closure) if and only if $F$ is equicontinuous and pointwise bounded.

The special case that is often taught in undergraduate analysis is when $X = [a,b]\subset\mathbb{R}$, $-\infty<a<b<\infty$.

Because $C(X)$ with supremum norm is a metric space, the notions of compactness, sequential compactness, and limit point compactness coincide. Therefore what Munkres refers to as Ascoli's theorem (which is about relative compactness of $F$) and Arzela's theorem (which is about relative sequential compactness of $F$) are in fact the same theorem, and most people do not make a distinction between the two.

Generalizations of this theorem may also be referred to as Ascoli-Arzela, or as an Ascoli-Arzela type theorem, depending on the author.


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