I've got the following question.
Let $\{\phi_n\}$ be any orthonormal basis of the reproducing kernel Hilbert space $H_0$. Let $F:H_0\to L^2$ be a linear isometry. Is $\xi_n=F(\phi_n)$ an orthonormal basis of $L^2$? And if yes, would this implie that $\xi_n$ are independent?
As $F$ is an isometry and $\phi_n$ is an orthonormla basis, I know that $\xi_n$ has to be an orthonormal system. But I couldn't find any theorem about it beeing a basis. And I'm not sure, if for random variable being a basis implies independence.
Thanks a lot!