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I’m wondering if there is a version of Arzelà-Ascoli for continuous functions on not-necessarily compact metric spaces $X$, i.e. a characterization of the compact subsets of $C_b(X)$ for $X$ not-necessarily compact, e.g. the titular $X=(0,1)$, using a neat set of conditions. I used to have $C(X)$ here instead of $C_b(X)$ but @EricWofsey's comment "simplified" it to $C_b(X)$, where the "sup norm" is actually a norm. Is a Closed and Bounded Interval Required for Arzela-Ascoli? tells us that the same set of conditions as the original Arzelà-Ascoli theorem do not hold for non-compact spaces, and Generalizations of Arzela Ascoli theorem brings up compact Hausdorff spaces.

I assume there is not such a characterization, which brings me to perhaps the more important question: is there a philosophical reason or guiding intuition/heuristic that tells us that we “should not expect” such a theorem for non-compact spaces $X$?

Or dually is there a philosophical reason or guiding intuition/heuristic that tells us that we “should expect” such a theorem for compact spaces $X$? I know that continuous functions on compact sets enjoy a wide variety of nice properties (boundedness, attaining extrema, uniformity of continuity and convergence, etc.) but how do I know that these properties are “the best properties” and therefore indicative that continuous functions “really should belong on compact sets”/“live most naturally on compact sets”?

(cross posted to MO: https://mathoverflow.net/questions/412637/arzel%c3%a0-ascoli-for-c-b0-1-or-more-generally-why-is-that-continuous-function)

EDIT: Classifying the compact subsets of $L^p$ tells us there is a nice characterization of compact subsets of $L^p(\mathbb R^n)$, and the Hanche-Olsen & Holden paper the answer links (https://arxiv.org/pdf/0906.4883.pdf) references an article of Phillips (Thm. 3.7 of https://www.ams.org/journals/tran/1940-048-03/S0002-9947-1940-0004094-3/) providing a characterization of compact subsets of any Banach space, including of course $C_b(X)$ for any topological space $X$. I am unsure as to the usefulness of Phillips' characterization.

D.R.
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  • What topology are you using on $C(X)$? The compact-open topology? – Eric Wofsey Dec 22 '21 at 04:21
  • @EricWofsey I was thinking just the sup norm topology. – D.R. Dec 22 '21 at 05:05
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    The sup norm need not be bounded on a non-compact space. The compact open topology generalizes this, so that convergence $f_n \to f$ in the compact open topology occurs if and only if those functions converge uniformly to $f$ on all compact subsets. – paul blart math cop Dec 22 '21 at 06:10
  • @paulblartmathcop can't we still talk about the topology "generated" by the sup norm on non-compact spaces? – D.R. Dec 22 '21 at 09:08
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    You can, but it will just be a disjoint union of the equivalence classes where two functions are equivalent if the distance between them is bounded. In other words, it will just be a huge disjoint union of copies of $C_b(X)$ (the space of bounded continuous functions on $X$ with the sup norm), so you might as well restrict your attention to $C_b(X)$ where you have a genuine norm. – Eric Wofsey Dec 22 '21 at 15:21
  • @EricWofsey ok, thank you for clearing up that case. Now, I suppose that "simplifies" my above question to just consider $C_b(X)$ for $X$ non-necessarily compact (still under the sup norm topology) – D.R. Dec 23 '21 at 04:06

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