GoI for MELL: exponentials - Revision history

Laurent Regnier: definition of type !A

2010-11-17T11:29:15Z

definition of type !A

Laurent Regnier: style

2010-11-17T10:55:50Z

style

Laurent Regnier: The copying iso

2010-06-05T18:26:05Z

The copying iso

Laurent Regnier: /* The tensor product of Hilbert spaces */ presentation

2010-06-05T10:20:44Z

‎The tensor product of Hilbert spaces: presentation

Laurent Regnier: Creation of the page : generalities on Hilbert spaces tensor product

2010-05-25T08:28:09Z

Creation of the page : generalities on Hilbert spaces tensor product

New page

= The tensor product of Hilbert spaces</math> =

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:
: <math>\langle (x_{np}), (y_{np})\rangle = \sum_{n,p} x_{np}\bar y_{np}</math>.

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:
: <math>e_{ijnp} = \delta_{in}\delta_{jp}</math> (all terms are null but the one at index <math>(i,j)</math> which is 1).

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:
: <math>x\tens y = (x_ny_p)_{n,p\in\mathbb{N}}</math>.

In particular we have: <math>e_{ij} = e_i\tens e_j</math> and we can write:
: <math>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}</math>

@@ Line 21: / Line 21: @@
 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.
-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
 {{Remark|
@@ Line 51: / Line 51: @@
 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:
 : <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>.
-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

@@ Line 1: / Line 1: @@
 = The tensor product of Hilbert spaces =
-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 we work in the Hilbert space <math>H=\ell^2(\mathbb{N})</math> endowed with its canonical hilbertian basis denoted by <math>(e_k)_{k\in\mathbb{N}}</math>.
 : <math>\langle (x_{np}), (y_{np})\rangle = \sum_{n,p} x_{np}\bar y_{np}</math>.
@@ Line 7: / Line 7: @@
 : <math>x\tens y = (x_ny_p)_{n,p\in\mathbb{N}}</math>.
-Recall that <math>(e_k)_{k\in\mathbb{N}}</math> is the canonical hilbertian basis of <math>H=\ell^2(\mathbb{N})</math>. We define: <math>e_{np} = e_n\tens e_p</math> so that <math>e_{np}</math> is the sequence <math>(e_{npij})_{i,j\in\mathbb{N}}</math> of complex numbers given by <math>e_{npij} = \delta_{ni}\delta_{pj}</math>. By bilinearity of tensor we have:
+We define: <math>e_{np} = e_n\tens e_p</math> so that <math>e_{np}</math> is the sequence <math>(e_{npij})_{i,j\in\mathbb{N}}</math> of complex numbers given by <math>e_{npij} = \delta_{ni}\delta_{pj}</math>. By bilinearity of tensor we have:
 : <math>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}</math>

@@ Line 1: / Line 1: @@
 = 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:
+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:
 : <math>\langle (x_{np}), (y_{np})\rangle = \sum_{n,p} x_{np}\bar y_{np}</math>.
@@ Line 7: / Line 7: @@
 : <math>x\tens y = (x_ny_p)_{n,p\in\mathbb{N}}</math>.
-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:
+Recall that <math>(e_k)_{k\in\mathbb{N}}</math> is the canonical hilbertian basis of <math>H=\ell^2(\mathbb{N})</math>. We define: <math>e_{np} = e_n\tens e_p</math> so that <math>e_{np}</math> is the sequence <math>(e_{npij})_{i,j\in\mathbb{N}}</math> of complex numbers given by <math>e_{npij} = \delta_{ni}\delta_{pj}</math>. By bilinearity of tensor we have:
 : <math>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}</math>

@@ Line 1: / Line 1: @@
-= 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:
 : <math>\langle (x_{np}), (y_{np})\rangle = \sum_{n,p} x_{np}\bar y_{np}</math>.
 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:
 : <math>x\tens y = (x_ny_p)_{n,p\in\mathbb{N}}</math>.
-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>.