# GoI for MELL: exponentials

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(Creation of the page : generalities on Hilbert spaces tensor product) |
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− | = The tensor product of Hilbert spaces</math> = |
+ | = The tensor product of Hilbert spaces = |

Recall that <math>(e_k)_{k\in\mathbb{N}}</math> is the canonical basis of <math>H=\ell^2(\mathbb{N})</math>. The space <math>H\tens H</math> is the collection of sequences <math>(x_{np})_{n,p\in\mathbb{N}}</math> of complex numbers such that: <math>\sum_{n,p}|x_{np}|^2</math> converges. The scalar product is defined just as before: |
Recall that <math>(e_k)_{k\in\mathbb{N}}</math> is the canonical basis of <math>H=\ell^2(\mathbb{N})</math>. The space <math>H\tens H</math> is the collection of sequences <math>(x_{np})_{n,p\in\mathbb{N}}</math> of complex numbers such that: <math>\sum_{n,p}|x_{np}|^2</math> converges. The scalar product is defined just as before: |
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: <math>\langle (x_{np}), (y_{np})\rangle = \sum_{n,p} x_{np}\bar y_{np}</math>. |
: <math>\langle (x_{np}), (y_{np})\rangle = \sum_{n,p} x_{np}\bar y_{np}</math>. |
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− | |||

− | The canonical basis of <math>H\tens H</math> is denoted <math>(e_{ij})_{i,j\in\mathbb{N}}</math> where <math>e_{ij}</math> is the (doubly indexed) sequence <math>(e_{ijnp})_{n,p\in\mathbb{N}}</math> defined by: |
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− | : <math>e_{ijnp} = \delta_{in}\delta_{jp}</math> (all terms are null but the one at index <math>(i,j)</math> which is 1). |
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If <math>x = (x_n)_{n\in\mathbb{N}}</math> and <math>y = (y_p)_{p\in\mathbb{N}}</math> are vectors in <math>H</math> then their tensor is the sequence: |
If <math>x = (x_n)_{n\in\mathbb{N}}</math> and <math>y = (y_p)_{p\in\mathbb{N}}</math> are vectors in <math>H</math> then their tensor is the sequence: |
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: <math>x\tens y = (x_ny_p)_{n,p\in\mathbb{N}}</math>. |
: <math>x\tens y = (x_ny_p)_{n,p\in\mathbb{N}}</math>. |
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− | In particular we have: <math>e_{ij} = e_i\tens e_j</math> and we can write: |
+ | In particular if we define: <math>e_{np} = e_n\tens e_p</math> so that <math>e_{np}</math> is the (doubly indexed) sequence of complex numbers given by <math>e_{npij} = \delta_{ni}\delta_{pj}</math> then <math>(e_{np})</math> is a hilbertian basis of <math>H\tens H</math>: the sequence <math>x=(x_{np})</math> may be written: |

− | : <math>x\tens y = \left(\sum_n x_ne_n\right)\left(\sum_p y_pe_p\right) = |
+ | : <math>x = \sum_{n,p\in\mathbb{N}}x_{np}\,e_{np} |

− | \sum_{n,p} x_ny_p e_n\tens e_p = \sum_{n,p} x_ny_p e_{np}</math> |
+ | = \sum_{n,p\in\mathbb{N}}x_{np}\,e_n\tens e_p</math>. |

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

## Revision as of 12:20, 5 June 2010

# 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: