  It should be stressed that at this point there is no direct connection between R3 and physical space. KS wish to show that for an arbitrary QM system requiring a representation in a Hilbert space of at least three dimensions, the ascription of values in conjunction with condition (KS2) (Sum Rule and Product Rule) is impossible, and in order to do this it is sufficient to consider the space R3. This space R3, however, does not represent physical space for the quantum system at issue. In particular, orthogonality in R3 is not to be confused with orthogonality in physical space. This becomes obvious if we move to an example of a QM system sitting in physical space and at the same time requiring a QM representation in H3, e.g. the spin degree of freedom of a one-particle spin-1 system.  Given an arbitrary direction α in physical space and an operator  Sα  representing the observable of a spin component in direction  α,  H3 is spanned by the eigenvectors of  Sα,  namely  |Sα=1&gt;,   |Sα=0&gt;,  |Sα=−1&gt;,  which are mutually orthogonal in H3. The fact that these three vectors corresponding to three possible results of measurement in one spatial direction are mutually orthogonal illustrates the different senses of orthogonality in H3 and in physical space. (The reason lies, of course, in the structure of QM, which represents different values of an observable by different directions in H3.) 