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Let $X$ and $Y$ be Banach spaces and let $C(X,Y)$ be the set of continuous functions from $X$ to $Y$ equipped with the topology of uniform convergence on compact sets (i.e. the compact-open topology). Consider the sub-collection $C^{\star}(X,Y)$ consisting of all $f\in C(X,Y)$ which are $\alpha$-Hölder continuous for some $\alpha \in (0,1]$.

Consider the two-part question:

  • Is $C^{\star}(X,Y)$ dense in $C(X,Y)$?
  • If so, is $C^{\star}(X,Y)$ generic (in the sense of Baire-Category; i.e. residual) in $C(X,Y)$ for the compact-open topology?

Thoughts: If $X$ and $Y$ both have the BAP, then the first point should be true (no idea about the second point), but I would prefer not to assume this.

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    $\begingroup$ C*(X,Y) is a countable union of closed sets with empty interior. (Also note that the compact open topology on C(X,Y) is not metrizable in general). $\endgroup$
    – Pietro Majer
    Jun 14, 2022 at 9:18
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    $\begingroup$ And if i'm not wrong, even "Lipschitz maps with range in finite dimensional subspaces" are a dense subset of C(X,Y) wrto the c.o. topology. $\endgroup$
    – Pietro Majer
    Jun 14, 2022 at 9:26
  • $\begingroup$ @PietroMajer Would you happen to have a reference to this last fact? $\endgroup$
    – ABIM
    Jun 14, 2022 at 10:52
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    $\begingroup$ You seem to ask many similar questions ("is P residual in Q" where in most cases the answer is strongly negative since the complement of P is residual). In any case here you have a subspace of a space. A proper subspace of a Hausdorff topological space is never residual, since it's disjoint to some of its translates. $\endgroup$
    – YCor
    Jun 14, 2022 at 11:06
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    $\begingroup$ You've been asking 4 questions in less than 24h mathoverflow.net/questions/424645/…, mathoverflow.net/questions/424670, now-deleted mathoverflow.net/questions/424630. I'd recommend to more thoroughly think about them before posting. $\endgroup$
    – YCor
    Jun 14, 2022 at 11:12

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