# GoI for MELL: exponentials

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# The tensor product of Hilbert spaces[/itex]

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:

$\langle (x_{np}), (y_{np})\rangle = \sum_{n,p} x_{np}\bar y_{np}$.

The canonical basis of $H\tens H$ is denoted $(e_{ij})_{i,j\in\mathbb{N}}$ where eij is the (doubly indexed) sequence $(e_{ijnp})_{n,p\in\mathbb{N}}$ defined by:

eijnp = δinδjp (all terms are null but the one at index (i,j) which is 1).

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 we have: $e_{ij} = e_i\tens e_j$ and we can write:

$x\tens y = \left(\sum_n x_ne_n\right)\left(\sum_p y_pe_p\right) = \sum_{n,p} x_ny_p e_n\tens e_p = \sum_{n,p} x_ny_p e_{np}$