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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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closed as unclear what you're asking by Yemon Choi, Joonas Ilmavirta, Alex Degtyarev, Ryan Budney, Johannes Hahn May 11 '15 at 17:57

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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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 May 10 '15 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 May 10 '15 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 May 10 '15 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 May 10 '15 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 May 11 '15 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 May 12 '15 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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