## Appendix A Hilbert Spaces With Denumerable Basis  
**Definition A1** (Hilbert space of configurations). : _Given a denumerable set \(\mathcal{B}\), with \(\ell^{2}(\mathcal{B})\) we shall denote the infinite dimensional Hilbert space defined as follow._  
_The set of vectors in \(\ell^{2}(\mathcal{B})\) is the set_  
\[\left\{\phi\mid\phi:\mathcal{B}\rightarrow\mathbb{C},\sum_{C\in\mathcal{B}}| \phi(C)|^{2}<\infty\right\}\]  
_and equipped with:_  
1. _An inner sum_ \(+:\ell^{2}(\mathcal{B})\times\ell^{2}(\mathcal{B})\rightarrow\ell^{2}( \mathcal{B})\) _defined by_ \((\phi+\psi)(C)=\phi(C)+\psi(C)\)_;_
2. _A multiplication by a scalar_ \(\dots:\mathbb{C}\times\ell^{2}(\mathcal{B})\rightarrow\ell^{2}(\mathcal{B})\) _defined by_ \((a\cdot\phi)(C)=a\cdot(\phi(C))\)_;_
3. _An inner product_ \((\cdot,\cdot):\ell^{2}(\mathcal{B})\times\ell^{2}(\mathcal{B})\rightarrow\mathbb{C}\) _defined by_ \(\langle\phi,\psi\rangle=\sum_{C\in\mathcal{B}}\phi(C)^{*}\psi(C)\) _(observe that the condition_ \(\sum_{C\in\mathcal{B}}|\phi(C)|^{2}<\infty\) _implies that_ \(\sum_{C\in\mathcal{B}}\phi(C)^{*}\psi(C)\) _converges for every pair of vectors)._
4. _The Euclidian norm is defined as_ \(\|\phi\|=\sqrt{\langle\phi,\phi\rangle}\)_._  
The Hilbert space \(\ell^{2}=\ell^{2}(\mathbb{N})\) is the standard Hilbert space of denumerable dimension--all the Hilbert spaces with denumerable dimension are isomorphic to it. \(\ell_{1}^{2}\) is the set of the vectors of \(\ell^{2}\) with unit norm. **Definition A2** (computational basis). : _The set of functions_  
\[\mathsf{CB}(\mathcal{B})=\left\{|C\rangle:C\in\mathcal{B},\;|C\rangle:\mathcal{ B}\rightarrow\mathbb{C}\right\}\]  
_such that for each \(C\)_  
\[|C\rangle(D)=\left\{ \begin{array}{ll}1&\mbox{if }C=D\\ 0&\mbox{if }C\
eq D\end{array} \right.\]  
_is called computational basis of \(\ell^{2}(\mathcal{B})\)._  
We can prove that [16]:  
**Theorem A1**. : _The set \(\mathsf{CB}(\mathcal{B})\) is a Hilbert basis of \(\ell^{2}(\mathcal{B})\)._  
Let us note that the inner product space \(\mathsf{span}(\mathsf{CB}(\mathcal{B}))\) defined by:  
\[\mathsf{span}(\mathsf{CB}(\mathcal{B}))=\left\{\sum_{i=1}^{n}c_{i}S_{i}\mid c _{i}\in\mathbb{C},S_{i}\in\mathsf{CB}(\mathcal{B}),n\in\mathbb{N}\right\}.\]  
is a proper inner product subspace of \(\ell^{2}(\mathcal{B})\), but it is not an Hilbert Space (this means that \(\mathsf{CB}(\mathcal{B})\) is not an Hamel basis of \(\ell^{2}(\mathcal{B})\)). The completion of \(\mathsf{span}(\mathsf{CB}(\mathcal{B}))\) is a space isomorphic to \(\ell^{2}(\mathcal{B})\). By means of a standard result in functional analysis we have:  
**Theorem A2**. :
1. \(\mathsf{span}(\mathsf{CB}(\mathcal{B}))\) _is a dense subspace of_ \(\ell^{2}(\mathcal{B})\)_;_
2. \(\ell^{2}(\mathcal{B})\) _is the (unique! up to isomorphism)_ completion _of_ \(\mathsf{span}(\mathsf{CB}(\mathcal{B}))\)_._  
**Definition A3**. : _Let \(\mathcal{V}\) be a complex inner product space, a linear application \(U:\mathcal{V}\to\mathcal{V}\) is called an isometry if \(\langle\left\langle{{Ux},{Uy}}\right\rangle=\langle{x},{y}\rangle\), for each \(x,y\in\mathcal{V}\); moreover if \(U\) is also surjective, then it is called unitary._  
Since an isometry is injective, a unitary operator is invertible, and moreover, its inverse is also unitary. **Definition A4**. : _Let \(\mathcal{V}\) be a complex inner product vector space, a linear application \(L:\mathcal{V}\to\mathcal{V}\) is called bounded if \(\exists c>0\)\(\forall x\)\(\|{Lx}\|\leq c\|{x}\|\)._  
**Theorem A3**. : _Let \(\mathcal{V}\) be a complex inner product vectorial space, for each bounded application \(U:\mathcal{V}\to\mathcal{V}\) there is one and only one bounded application \(U^{*}:\mathcal{V}\to\mathcal{V}\) s.t. \(\langle{x},{Uy}\rangle=\langle{U^{*}x},{y}\rangle\). We say that \(U^{*}\) is the adjoint of \(U\)._  
It is easy to show that if \(U\) is a bounded application, then \(U\) is unitary iff \(U\) is invertible and \(U^{*}=U^{-1}\). **Theorem A4**.