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I've been reading about RKHSs and Hilbert spaces of functions these days a bit these days and I haven't yet come across an example of a hilbert space $H$ whose elements are all functions $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$ for $n,m>1$ and for which the evaluation functions $E_x:f\mapsto f(x)$ are bounded.

Do such objects exist and if so what are some well-known examples?

The only thing I have at the moment is the space of $n\times m$ matrices with Frobenius norm...which is a bit underwhelming...

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    $\begingroup$ What is your definition of a RKHS? Because wikipedia thinks a RKHS is a Hilbert space of real/complex-valued functions on a set. So I don't know how one could consider functions to $\mathbb R^m$; nor do I see how matrices fit into this framework, unless we think of matrices as functions from $[n^2]$ to $\mathbb C$? $\endgroup$
    – Matthew Daws
    Commented Sep 11, 2020 at 12:07
  • $\begingroup$ Oh for me its a hilbert space whose elements are functions (not to the base-field but to a topological vector space) and whose evaluation map is bounded linear operator $\endgroup$
    – ABIM
    Commented Sep 11, 2020 at 12:30
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    $\begingroup$ What about Sobolev spaces $H^s(\mathbb{R}^n; \mathbb{R}^m)$ for $s > n/2$? I thought this was kind of the canonical example. $\endgroup$
    – Nate Eldredge
    Commented Sep 11, 2020 at 13:42
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    $\begingroup$ Maybe I'm missing something, but why would it not work? Each coordinate function $f^{(i)}$ is $H^s(\mathbb{R}^n)$ and so it's continuous and its evaluation functions are bounded, and thus the evaluation functions for $f$ should likewise be bounded with a norm $\sqrt{m}$ times larger. $\endgroup$
    – Nate Eldredge
    Commented Sep 11, 2020 at 13:52
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    $\begingroup$ The standard spaces of entire functions (de Branges space, or just Paley-Wiener space) give you $m=n=2$, and for higher values, you can just take orthogonal sums of these and interpret them as functions on $\mathbb C^N$. $\endgroup$
    – Christian Remling
    Commented Sep 11, 2020 at 14:00

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