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Let $C(X)$ be the space of continuous functions with the usual norm. $X$ be a compact metric space. The Arzela Ascolis theorem says:

A subset $S$ of $C(X)$ is compact iff it is uniformly bounded and equicontinuous at any point of $x$.

I think the uniform boundedness can be relaxed to that $S$ is bounded at any point $x$.

Is that actually true or are there prominent counter examples?

MackeyTopology
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  • If you relax it to pointwise bounded then you get precompactness instead of compactness – Lorago Dec 16 '23 at 22:37
  • I see thank you! – MackeyTopology Dec 16 '23 at 22:39
  • @Lorago are you sure? I think if we not assume $F$ to be closed, we get always just precompactness, no? And uniform boundedness is just unecessary I think because picking a subsequence we only need boundedness of the whole set for any $t$. – MackeyTopology Dec 17 '23 at 02:21
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    You can check the wikipedia page on it, where one of the statements is "Let $X$ be a compact Hausdorff space. Then a subset $F$ of $C(X)$ is relatively compact in the topology induced by the uniform norm if and only if it is equicontinuous and pointwise bounded." – Lorago Dec 17 '23 at 07:51

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