# GoI for MELL: exponentials

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# The tensor product of Hilbert spaces

Recall that is the canonical basis of . The space is the collection of sequences of complex numbers such that:

∑ | | x_{np} | ^{2} |

n,p |

converges. The scalar product is defined just as before:

- .

If and are vectors in *H* then their tensor is the sequence:

- .

In particular if we define: so that *e*_{np} is the (doubly indexed) sequence of complex numbers given by *e*_{npij} = δ_{ni}δ_{pj} then (*e*_{np}) is a hilbertian basis of : the sequence *x* = (*x*_{np}) may be written:

- .

By bilinearity of tensor we have: