Appendix

Abstract

In this section we briefly recall the basics of Hilbert space theory. Although this will not be necessary for the typical reader it gives us the opportunity to fix notation. Consider a complex vector space “ endowed with a scalar product $( \cdot | \cdot )$. By the latter we mean a positive definite sesquilinear form $( \cdot | \cdot ):\mathcal{H} \times \mathcal{H} \to \mathbb{C}$, i.e., a mapping which is linear in the first variable, conjugate linear in the second variable,

$$\begin{array}{*{20}{c}} {\forall x,y,z \in \mathcal{H},\lambda ,\mu \in \mathbb{C}:} \\ {(\lambda \cdot x + \mu \cdot y|z) = \lambda (x|z) + \mu (y|z),(z|\lambda \cdot x + \mu \cdot y) = \bar{\lambda }(z|x) + \bar{\mu }(z|y),} \\ {(y|x) = \overline {(x|y)} } \\ \end{array}$$

and

$$(x|x) > 0 whenever x \ne 0.$$