# The tensor product of Hilbert spaces

Recall that $(e_k)_{k\in\mathbb{N}}$ is the canonical basis of $H=\ell^2(\mathbb{N})$. The space $H\tens H$ is the collection of sequences $(x_{np})_{n,p\in\mathbb{N}}$ of complex numbers such that:

 ∑ | xnp | 2 n,p

converges. The scalar product is defined just as before: $\langle (x_{np}), (y_{np})\rangle = \sum_{n,p} x_{np}\bar y_{np}$.

If $x = (x_n)_{n\in\mathbb{N}}$ and $y = (y_p)_{p\in\mathbb{N}}$ are vectors in H then their tensor is the sequence: $x\tens y = (x_ny_p)_{n,p\in\mathbb{N}}$.

In particular if we define: $e_{np} = e_n\tens e_p$ so that enp is the (doubly indexed) sequence of complex numbers given by enpij = δniδpj then (enp) is a hilbertian basis of $H\tens H$: the sequence x = (xnp) may be written: $x = \sum_{n,p\in\mathbb{N}}x_{np}\,e_{np} = \sum_{n,p\in\mathbb{N}}x_{np}\,e_n\tens e_p$.

By bilinearity of tensor we have: $x\tens y = \left(\sum_n x_ne_n\right)\tens\left(\sum_p y_pe_p\right) = \sum_{n,p} x_ny_p\, e_n\tens e_p = \sum_{n,p} x_ny_p\,e_{np}$