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Let $\mathcal{H}$ be a Hilbert space with inner product $\langle \cdot, \cdot \rangle$.

A priori, we do not know if the objects in this space are functions, numbers or something else entirely. How can I check if this space is an RKHS (Reproducing Kernel Hilbert Space)? Do I need some notion of a kernel or an input space a priori or do I have sufficient information already in the Hilbert space to deduce whether it could be an RKHS or not? Can we in some sense deduce whether the elements of $\mathcal{H}$ are functions?

Maybe this question is ill-posed but I find it odd that RKHS's appear so common and yet every definition starts off with the input space given. I'm just trying to figure out where the regularity of the RKHS requirement is posed, is it in the structure of the Hilbert space or the interplay between the Hilbert space and the input space?

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  • $\begingroup$ Sorry, what does 'RKHS' stand for? $\endgroup$ – Berci Feb 17 at 11:47
  • $\begingroup$ @Berci Reproducing Kernel Hilbert Space (edited) $\endgroup$ – Lundborg Feb 17 at 11:49
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$\mathcal{H}$ needs to be a space of functions from $X \to \mathbb{R}$ for it to be an RKHS. There are generalizations replacing $\mathbb{R}$ with any vector space.

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