An identity on direct sum of Hilbert spaces

Question

Let $M_i$ are the set of smooth complex valued functions ($i=0,1,2,...$) $L^2(M_i)$ are Hilbert spaces on $M_i$ then can we say $$L^2(\bigoplus_{i=0}^\infty M_i)\cong \bigoplus_{i=0}^\infty L^2(M_i)$$? Is there any refference for proof of this identity ? this identity is correct?

Answer 1

This is correct with the right interpretation of the symbol $\oplus$. On the left it should denote the disjoint union, on the right the Hilbert space sum. You should also be more explicit about the measures you are using (perhaps the spaces are Riemannian manifolds with their natural measures).