I have been studying Reproducing Kernel Hilbert Spaces (RKHS). The definition I am using is as follows: An RKHS is a Hilbert space $\mathcal{H}$ of real-valued functions on a set $X$ such that for all $x \in X$, the evaluation functional $E_x : \mathcal{H} \to \mathbb{R}$ defined by $E_x(f) = f(x)$ is continuous. (Recall that continuity in this context means $\sup_{\|f\|\leq 1}|f(x)| < \infty$ for all $x$.)
As one of the first steps in my study I want to see some positive and negative examples of these spaces. So I want to find a non-trivial (meaning infinite dimensional) space of functions which is a Hilbert space but not an RKHS, meaning the evaluation functionals are not continuous.
The problem is that in all the references I find, the only example they give is the space of functions $L_{2}$. However, this example is trivial since $L_2$ fails for the simple reason that it is not a space of functions, rather it contains equivalence classes of functions.
Can someone please help me find an infinite dimensional Hilbert space of functions that is not an RKHS?