Is every Hilbert space an $L^2$ space?

Question

Let $H$ be any Hilbert space. Must there exist a measure space $(X,\scr{M},\mu)$ such that we have a Hilbert space isomorphism: $$H\cong L^2(\mu)$$

Thank you

Answer 1

Yes. If $\kappa>0$ is the (Hilbert) dimension of $H$, then you can take for $\mu$ the counting measure on $\kappa$ and then it's easy to see that $H\cong L^2(\mu)=\ell^2(\kappa)$.

If $\kappa=0$, then take any $\mu$ such that $\mu(X)=\infty$ for any nonempty set $X$. Then $L^2(\mu)=\{0\}$.