Example of an infinite dimensional Hilbert space that is not an RKHS

Question

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?

Answer 1

There are no explicit examples of this kind. Indeed, if one could show you such a space $\mathcal H$, then point evaluation $f\mapsto f(x_0)$ would give an explicit example of a discontinuous linear functional on a Hilbert space. And there can be no explicit examples of this kind, because it's consistent with ZF that they do not exist. Only with the Axiom of Choice can one give a (non-constructive) proof of their existence. This is discussed in many places, such as

• Discontinuous linear functional
• Unbounded linear operator defined on $l^2$

So, any time someone gives you a concrete Hilbert space of functions, you can be sure that it's either RKHS, or is not really a space of functions.