Thank you, this may be helpful.

Actually, I like this one. Even if universal quantification is implicit, it is better not to forget that it is there.

You probably know about this preprint, but some readers may not [ arxiv.org/abs/1802.09936 ]

Even this is not so ``natural'' a choice. What is the big reason to take all of this terms with weight one? (It is not invariant with respect to scaling, for example.) I admit that in this case the question actually has an answer, but I also admit that, personally, I totally would not care about it.

I believe that this is not a meaningful question. You see, even though $H^s(\Omega)$ is called a Hilbert space it is technically not, in the sense that the actual Hilbert norm on it is not fixed. It is a sort of infinite dimensional analog of a linear space of finite dimension: you are free to chose any of the (equivalent) scalar products. Naturally, an orthonormal basis would very much depend on the choice.

I do not mind, and I do not have any specific requests.

Yes, but I had a hope for something more explicit.