 As stressed by von Neumann, the \(\{0,1\}\)-valued observables may be regarded as encoding propositions about—or, to use his phrasing, properties of—the state of the system. It is not difficult to show that a self-adjoint operator \(P\) with spectrum contained in the two-point set \(\{0,1\}\) must be a projection; i.e., \(P^2 = P\). Such operators are in one-to-one correspondence with the closed subspaces of \(\mathbf{H}\). Indeed, if \(P\) is a projection, its range is closed, and any closed subspace is the range of a unique projection. If \(u\) is any unit vector, then \(\langle Pu,u\rangle = \llvert Pu\rrvert ^2\) is the expected value of the corresponding observable in the state represented by \(u\). Since this observable is \(\{0,1\}\)-valued, we can interpret this expected value as the probability that a measurement of the observable will produce the “affirmative” answer 1. In particular, the affirmative answer will have probability 1 if and only if Pu = u; that is, \(u\) lies in the range of \(P\). Von Neumann concludes that 