It is a known fact that any two classical Hilbert spaces are isomorphic, in particular, any classical Hilbert space is isomorphic to $L^2$.
Consider now weighted Hilbert spaces, that is, Hilbert spaces with a weighted inner product
$ \langle f, g \rangle = \int f \ g \ w \ dx $
where $w(x)$ is the weight function. This situation appears frequently in Sturm-Liouville eigenvalue problems. Let's call $H_w$ to such weighted Hilbert space.
My question is: are all weighted Hilbert spaces also isomorphic?. In other words, is any weighted Hilbert space $H_w$ isomorphic to $L^2_w$?
Thank you very much.