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 $f:mathbb{R}^nrightarrow mathbb{R}^m$ for $n,m>1$ and for which the evaluation functions $E_x:fmapsto f(x)$ 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 $ntimes m$ matrices with Frobenius norm…which is a bit underwhelming…