GoI for MELL: exponentials

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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

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}
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