Completeness criteria for inner product spaces

Abstract

When in 1933. A. N. Kolmogorov [171 first axiomatized contemporary probability theory, the internal structure of the set of experimentally verifiable assertions ( L) assigned to a physical system was mathematically envisaged as a Boolean algebra. The order in L corresponds to the relation of implication. and the lattice operations/\. meet (infimum) v. join (supremum). and the operation orthocom implementation correspond to the operations of conjunction. disjunction and complementation. A Boolean algebra satisfies the distributive law. i.e. for any three elements a. b. c of it. we have a/\ (b V c) =(a/\ b) V (a/\ c). Moreover. by Stone's theorem (see for example: [25]). every Boolean algebra can be identified with an algebra of subsets of some non-empty set Q. In spite of its fruitfulness in describing classical systems. Boolean algebras result to be insufficient when it comes to quantum systems.