Tensor product of Hilbert spaces

Remarks

If $(e_i)_{i \in I}$ and $(f_j)_{j \in J}$ are orthonormal bases of $H_1$ and $H_2$, then $(e_i \otimes f_j)_{(i, j) \in I \times J}$ is an orthonormal basis of $H_1 \mathbin{\widehat{\otimes}} H_2$.

Example: $\ell^2(I) \mathbin{\widehat{\otimes}} \ell^2(J)$ is unitarily equivalent to $\ell^2(I \times J)$, by $e_i \otimes f_j \mapsto e_{(i, j)}$. Both sides have orthonormal bases indexed by $I \times J$, and the map sends one onto the other. Similarly, for $\sigma$-finite measure spaces, $L^2(X) \mathbin{\widehat{\otimes}} L^2(Y)$ is $L^2(X \times Y)$, with $f \otimes g$ sent to $(x, y) \mapsto f(x) g(y)$.

Non-example: if $H_1$ and $H_2$ are infinite-dimensional, the algebraic tensor product $H_1 \otimes H_2$ with the inner product above is not complete. For instance, $\sum_{n \ge 1} 2^{-n} e_n \otimes f_n$ converges in the completion but is not a finite sum of elementary tensors. In finite dimension no completion is needed, and $H_1 \mathbin{\widehat{\otimes}} H_2 = H_1 \otimes H_2$.

Other completions of $H_1 \otimes H_2$, for instance the projective tensor product, are Banach spaces with different norms.