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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?
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