# Reproducing kernel Hilbert sapce

I encountered the following claim (verbatim):

Theorem Let $V$ be a subspace of $L^2(\mathbb{R})$ and $\{e_n\}$ be a orthonormal basis of $V$. The $V$ is a reproducing kernel Hilbert space with kernel

$$K(x,y) = \sum_n e_n(x)e_n(y).$$

For any function $f \in V$,

$$f(y) = \int f(x) K(x,y) dy.$$

Questions:

• I am under the impression that a RKHS is a Hilbert space of functions, where pointwise evaluation is a bounded functional. $L^2$ is not even a space of functions, how does the claim make sense (e.g. take an ONB for the entire $L^2$)?
• Also, are there no convergence requirements on the reproducing kernel $K$, or is it purely formal?
-