How to verify $H\otimes K \cong \bigoplus\limits_{i\in I}H$

Question

Let $H,~K$ be the Hilbert space. if $\{v_{j}\}_{j\in J}\subset H$ and $\{w_{i}\}_{i\in I}\subset K$ are the orthonormal bases, then how to construct the isomorphic mapping: $H\otimes K \rightarrow \bigoplus\limits_{i\in I}H$. Here, the $H \otimes K$ is the tensor product of two Hilbert space.

Answer 1

The desired isomorphism $I:H\otimes K\to\oplus_{i\in I}H$ is well defined by $$ I(v_j\otimes w_i)_i=v_j $$ The rest is an easy exercise.