# GoI for MELL: exponentials

m (style) |
(definition of type !A) |
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Being both separable Hilbert spaces, <math>H</math> and <math>H\tens H</math> are isomorphic. We will now define explicitely an iso based on partial permutations. |
Being both separable Hilbert spaces, <math>H</math> and <math>H\tens H</math> are isomorphic. We will now define explicitely an iso based on partial permutations. |
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− | We fix, once for all, a bijection from couples of natural numbers to natural numbers that we will denote by <math>(n,p)\mapsto\langle n,p\rangle</math>. For example set <math>\langle n,p\rangle = 2^n(2p+1) - 1</math>. |
+ | We fix, once for all, a bijection from couples of natural numbers to natural |

+ | numbers that we will denote by <math>(n,p)\mapsto\langle n,p\rangle</math>. For |
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+ | example set <math>\langle n,p\rangle = 2^n(2p+1) - 1</math>. Conversely, given |
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+ | <math>n\in\mathbb{N}</math> we denote by <math>n_{(1)}</math> and |
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+ | <math>n_{(2)}</math> the unique integers such that <math>\langle n_{(1)}, |
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+ | n_{(2)}\rangle = n</math>. |
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{{Remark| |
{{Remark| |
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If we suppose that <math>u = u_\varphi</math> is a <math>p</math>-isometry generated by the partial permutation <math>\varphi</math> then we have: |
If we suppose that <math>u = u_\varphi</math> is a <math>p</math>-isometry generated by the partial permutation <math>\varphi</math> then we have: |
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: <math>!u(e_{\langle n,p\rangle}) = \Phi(e_n\tens u(e_p)) = \Phi(e_n\tens e_{\varphi(p)}) = e_{\langle n,\varphi(p)\rangle}</math>. |
: <math>!u(e_{\langle n,p\rangle}) = \Phi(e_n\tens u(e_p)) = \Phi(e_n\tens e_{\varphi(p)}) = e_{\langle n,\varphi(p)\rangle}</math>. |
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− | Thus <math>!u</math> is itself a <math>p</math>-isometry and the proof space is stable under the copying iso. |
+ | Thus <math>!u_\varphi</math> is itself a <math>p</math>-isometry generated by the |

+ | partial permutation <math>!\varphi:n\mapsto \langle n_{(1)}, \varphi(n_{(2)})\rangle</math>, which shows that the proof space is stable under the copying iso. |
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+ | |||

+ | Given a type <math>A</math> we define the type <math>!A</math> by: |
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+ | : <math>!A = \{!u, u\in A\}\biorth</math> |

## Latest revision as of 13:29, 17 November 2010

# [edit] The tensor product of Hilbert spaces

Recall that we work in the Hilbert space endowed with its canonical hilbertian basis denoted by .

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:

- .

We define: so that *e*_{np} is the sequence of complex numbers given by *e*_{npij} = δ_{ni}δ_{pj}. By bilinearity of tensor we have:

Furthermore the system of vectors (*e*_{np}) is a hilbertian basis of : the sequence may be written:

- .

## [edit] An algebra isomorphism

Being both separable Hilbert spaces, *H* and are isomorphic. We will now define explicitely an iso based on partial permutations.

We fix, once for all, a bijection from couples of natural numbers to natural
numbers that we will denote by . For
example set . Conversely, given
we denote by *n*_{(1)} and
*n*_{(2)} the unique integers such that .

*Remark:*
just as it was convenient but actually not necessary to choose *p* and *q* so that *p**p*^{ * } + *q**q*^{ * } = 1 it is actually not necessary to have a *bijection*, a one-to-one mapping from *into* would be sufficient for our purpose.

This bijection can be extended into a Hilbert space isomorphism by defining:

- .

Now given an operator *u* on *H* we define the operator !*u* on *H* by:

- .

*Remark:*
The operator !*u* is defined by:

where denotes the operator on defined by for any *x*,*y* in *H*. However this notation must not be confused with the tensor of operators that was defined in the previous section in order to interpret the tensor rule of linear logic; we therefore will not use it.

One can check that given two operators *u* and *v* we have:

- !
*u*!*v*= !(*u**v*); - !(
*u*^{ * }) = (!*u*)^{ * }.

Due to the fact that Φ is an isomorphism *onto* we also have !1 = 1; this however will not be used.

We therefore have that ! is a morphism on ; it is easily seen to be an iso (not *onto* though). As this is the crucial ingredient for interpreting the structural rules of linear logic, we will call it the *copying iso*.

## [edit] Interpretation of exponentials

If we suppose that is a *p*-isometry generated by the partial permutation then we have:

- .

Thus is itself a *p*-isometry generated by the
partial permutation , which shows that the proof space is stable under the copying iso.

Given a type *A* we define the type !*A* by: