# Is Arzela-Ascoli with equicontiuous or uniformly equicontinuous

I am still working on this proof of Arzela-Ascoli but now I noticed that in my statement of the theorem I used ''equicontinuous'' to mean ''uniformly equicontinuous''. At least in the direction I managed I did not find the mistake this should cause. Hence I am now wondering:

Are both of the following true theorems?

Let $X$ be a compact Hausdorff space and let $C(X)$ denote the space of continuous functions $f: X \to \mathbb R$ endowed with the sup norm $\|\cdot \|_\infty$. Then $S \subseteq C(X)$ is relatively compact if and only if it is pointwise bounded and equicontinuous.

Let $X$ be a compact Hausdorff space and let $C(X)$ denote the space of continuous functions $f: X \to \mathbb R$ endowed with the sup norm $\|\cdot \|_\infty$. Then $S \subseteq C(X)$ is relatively compact if and only if it is pointwise bounded and uniformly equicontinuous.

• With a compact domain, equicontinuity and uniform equicontinuity are equivalent. See here. – David Mitra Apr 18 '14 at 12:50
• @DavidMitra Thank you, now it's clear. – Student Apr 18 '14 at 12:51
• Perhaps it is also worth noting that most books (that I've seen) require that $S$ be uniformly bounded, but this too, is equivalent to pointwise boundedness when $X$ is compact. – nigel Apr 18 '14 at 13:21