**EDIT**:*Simple misunderstanding of statements like "$L_2$ is not a RKHS". Such statements do not make sense without reference to a kernel. In other words, a RKHS is a space*

__along with__a kernel..."I am a little confused about reproducing kernel Hilbert spaces, particularly the example that $L_2$ is not a RKHS because the delta function is not in $L_2$.

From the definition, it looks like to me like you can choose the Hilbert space and the reproducing kernel independently. Ok, so I consider $\mathcal{H}=L_2$ and the kernel $k$ to be the delta function. The example goes on to say that the reproducing property is o.k. because $f(x) = \int \delta(x-u)f(u)du$ but that the delta function is not in $L_2$, therefore $L_2$ is not a RKHS.

My understanding, which follows, seems to describe the above example. (What am I missing?) It seems I can start with any Hilbert space, and then I could just choose a kernel independently which is not in the Hilbert space and then declare that the Hilbert space is not a RKHS.

Thanks,

Gary