GoI for MELL: exponentials

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The tensor product of Hilbert spaces</math>

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

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}
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