Assume I have a pair of finite-dimensional vector spaces $V_A$ and $V_B$, and let $\mathcal{H}$ be their tensor-product space: $$ \mathcal{H}=V_A\otimes V_B. $$ Let $\langle \cdot , \cdot \rangle$ be an inner-product on $\mathcal{H}$.
My question is as follows: Do we have a Schmidt decomposition for any $v\in \mathcal{H}$. i.e. can we always have a representation $$ v=\sum_i \lambda_i \phi_i \otimes \psi_i , $$ where $\{\phi_i \otimes \psi_i\}_i\subset \mathcal{H}$ is an orthonormal set and $\lambda_i>0$, $\sum_i \lambda_i^2=1$?
In particular, does any inner-product allows an orthonormal basis of product vectors?