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Consider a space of (generalized) functions $F$ defined on a measure space $\Omega$ and equipped with a topology.

What are necessary and sufficient conditions for point evaluations at arbitrary $x \in \Omega$ to be well-defined and continuous on $F$?

Motivation: The Sobolev embedding theorem states that the Sobolev space $W^{k,p}(\mathbb R^n)$ embeds into the Hölder space $C^{r,\alpha}(\mathbb R^n)$ whenever $(k-r-\alpha)/n=1/p$. However, Hölder continuity is much stronger than continuity, which in turn is stronger than the existence of point evaluations. So I wonder whether those embeddings are really the most general you can get.

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    $\begingroup$ Point evaluarions are well-defined and continuous if and only if they are well-defined and continuous. I appreciate this is not the kind of answer you want; but at the level of generality you have adopted, the question feels a bit like a fishing expedition. What counts as a "condition"? $\endgroup$
    – Yemon Choi
    Commented May 10, 2015 at 23:18
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    $\begingroup$ It seems that before seeking "necessary and sufficient conditions", you should seek a precise definition of what it even means to have point evaluations for "generalized functions" or functions that are probably a priori defined only up to null sets. $\endgroup$
    – Eric Wofsey
    Commented May 10, 2015 at 23:18
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    $\begingroup$ The notion of "generalized function" that I am vaguely familiar with is in the context of distributions, which therefore requires a lot more structure than just having a measure space. $\endgroup$
    – Yemon Choi
    Commented May 10, 2015 at 23:19
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    $\begingroup$ Perhaps you are interested in multiplier algebras of RKHS? If so, then this should be specified in the question $\endgroup$
    – Yemon Choi
    Commented May 10, 2015 at 23:20
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    $\begingroup$ I have voted to close. Despite Paul Garrett's attempt below, I think the question is ill-posed in its present form. "Give necessary and sufficient conclusions for a property P" does not strike me as well-defined, and I think the OP should rethink what exactly he or she is looking for beyond "P holds iff P holds", before we get more answers trying to guess at what the OP wants $\endgroup$
    – Yemon Choi
    Commented May 11, 2015 at 13:32

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If your function space is a Hilbert space $H$ (and $\Omega\subset \mathbb R^n$) then point evaluations are continuous linear functionals if and only if the space is a reproducing kernel Hilbert space: $f(x) = ev_x(f)=\langle K_x, f\rangle$ for a unique element $K_x\in H$ and $K(x,y)=K_x(y) = ev_y(K_x) =\langle K_y,K_x\rangle$, etc.

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  • $\begingroup$ Doesn't this just restate the definition of a RKHS? If I understand correctly , a RKHS is the dual of a Hilbert space $H$ where a set $\Omega$ and an injection $\Omega \to H$ have been specified. $\endgroup$
    – Yemon Choi
    Commented May 12, 2015 at 11:31
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I'd think that a "measure space" is by far too weak a hypothesis on the underlying "physical space". E.g., continuity has no sense. So, granting at least that the space on which the functions exist is a topological space, with a (regular?) Borel measure, so that point-wise sense is arguably inarguable, you could take the (quasi-?) completion of continuous, compactly-supported continuous functions with respect to the topology given by the seminorms "evaluate at $x_o$" as $x_o$ ranges over the physical space.

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