I have heard this phrase quite a lot now, that RKHS is dense in the space of bounded continuous functions ($\mathcal L_2$).
For example, this would be true with
$$f(x) = \sum_{i=1}^N \alpha_i K(x_i,x)$$
for some $\alpha_i$, and $K$ the RBF kernel
$$K(x,y) = \text{exp}\left(-\frac{\|x-y\|_2^2}{\sigma}\right)$$
I guess this also means for any $x_i$ as well, or for infinite $N$. But which theorem / proof is this referring to?
Some options:
Mercer's representation theorem: $K(x,y)$ can be written as a linear combination of $\phi(x)\phi(y)$ where $\phi$ are the eigenfunctions of $K$
Riesz representer theorem: If $H$ is an RKHS, there exists a positive definite kernel function $K$ in which all functions in $H$ can be written this way.
Moore–Aronszajn: Any positive definite function $K$ defines an RKHS.
None of these quite seem like what I need though...
Edit: Here is a promising cuprit! The nonparametric representor theorem in
Scholkopf, Herbrich, and Smola: A Generalized Representer Theorem (2001)
The nonparametric version gives the functional class
$$\mathcal F = \left\{f | f(x) = \sum_{i=1}^\infty \beta_i k(x,z_i), \beta_i\in \mathbb R, z_i \in \mathcal X \subseteq \mathbb R^n, \|f\|< \infty \right\}$$
Then any $f\in \mathcal F$ minimizing some risk function with regularization $g(\|f\|)$ (for any monotonic $g$) has a representation
$$f(x) = \sum_{i=1}^n \alpha_i K(x_i,x).$$
However, it seems a nontrivial extension is needed to show that $\mathcal F$ is dense in $\mathcal L_2$.