  This manuscript addresses the problem of approximating an unknown function
from point evaluations. When obtaining these point evaluations is costly,
minimising the required sample size becomes crucial, and it is unreasonable to
reserve a sufficiently large test sample for estimating the approximation
accuracy. Therefore, an approximation with a certified quasi-optimality factor
is required. This article shows that such an approximation can be obtained when
the sought function lies in a \emph{reproducing kernel Hilbert space} (RKHS)
and is to be approximated in a finite-dimensional linear subspace. However,
selecting the sample points to minimise the quasi-optimality factor requires
optimising over an infinite set of points and computing exact inner products in
RKHS, which is often infeasible in practice. Extending results from optimal
sampling for $L^2$ approximation, the present manuscript proves that random
points, drawn independently from the Christoffel sampling distribution
associated with $\mcal{V}_d$, can yield a controllable quasi-optimality factor
with high probability. Inspired by this result, a novel sampling scheme, coined
subspace-informed volume sampling, is introduced and evaluated in numerical
experiments, where it outperforms classical i.i.d.\ Christoffel sampling and
continuous volume sampling. To reduce the size of such a random sample, an
additional greedy subsampling scheme with provable suboptimality bounds is
introduced. Our presentation is of independent interest to the community
researching the \emph{parametrised background data weak} (PBDW) method, as it
offers a simpler interpretation of the method.
