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From Wikipedia, two generalizations of the Arzelà–Ascoli theorem are

  1. Let $X$ be a compact Hausdorff space. Then a subset $F$ of $C(X)$, the set of real-valued continuous functions on X, is relatively compact in the topology induced by the uniform norm if and only if it is equicontinuous and pointwise bounded.
  2. Let $X$ be a compact Hausdorff space and $Y$ a metric space. Then a subset $F$ of $C(X,Y)$ is compact in the compact-open topology if and only if it is equicontinuous, pointwise relatively compact and closed.

$C(X)$ is a normed space with the uniform norm, and also a metric space under the metric induced by the uniform norm. I was wondering if the Arzelà–Ascoli theorem generalizations are direct results of applying to $C(X)$ some similar theorem(s) on normed spaces, metric spaces or other spaces (which $C(X)$ belongs to) ?

For example, are the Arzelà–Ascoli theorem generalizations results of the following theorem:

A subset in a metric space is compact iff it is complete and totally bounded?

Or the Arzelà–Ascoli theorem generalizations are not direct results of any similar theorems on normed spaces, metric spaces or other spaces (which $C(X)$ belongs to) ?

Thanks and regards!

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In books that I've read this generalizations was proved by checking completeness and total boundness. –  userNaN Feb 6 '12 at 0:01
    
@Norbert: Thanks! (1) May I have some references? (2) I guess you are talking about the second generalization. For the first generalization, it is not compactness but relative compactness. –  Tim Feb 6 '12 at 0:04
    
This is book in Russiаn Элементы теории функций и функциональный анализ. А. Н. Колмогоров, С. В. Фомин –  userNaN Feb 6 '12 at 0:06
    
The idea of the proofs of generalized theorems is the same as in thorem for $C([a,b])$. –  userNaN Feb 6 '12 at 0:07
    
@Norbert: Thanks! I found the English version "Elements of Function Theory and Functional Analysis by Kolmogorov and Fomin". Which chapter and volume? –  Tim Feb 6 '12 at 0:10

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