@user27182: link doesn't work, do you have an update?

@WillSawin Yes this gives a covariance function cov(X_s,X_t)=1/6-|s-t|(1-|s-t|).

Take P(x) the uniform measure on $\{x_1,\ldots,x_n\}$. Then you get roughly the usual construction of an RKHS...

The closure of such a subspace is $\mathcal H_k$ itself, i.e., it is a pre-Hilbert space.

@SandeepSilwal: the two inequalities are given in equations (2) and (11) in a survey by Fulton arxiv.org/abs/math/9908012

Many thanks! Just two queries if I may. For the fourth equality, so if $\cal F$ is a Banach space and $\cal R$ an RKHS, then $\text{clo}_{\cal F}\circ \text{clo}_{\cal R}=\text{clo}_{\cal F}$? Similarly the sixth equality, this is valid because the closure of the tensor product of two spaces is the closure of the tensor product of their closures?

Thanks for the advice, will do!

Thanks, that seems to be the right direction (support of a GP is the closure of its RKHS), but I do not intuit this at all yet. According to Lukic and Beder (ams.org/journals/tran/2001-353-10/S0002-9947-01-02852-5/…), in the infinite dimensional case the probability that a sample path of a GP lies in the corresponding RKHS is zero. So some definitions of support of the GP (say, roughly speaking, the smallest set such that sample paths are in it with probability one) should exclude the RKHS I guess...