The GNS construction validates the use of these spaces because it derives them as representations of positive forms on a C-star algebra. Symmetric or antisymmetric tensor products of these spaces yield a Fock space. A function from one of them to itself need only be surjective and norm-preserving to be a linear unitary operator, by Wigner's theorem
The GNS construction validates the use of these spaces because it derives them as representations of positive forms on a C-star algebra. Symmetric or antisymmetric tensor products of these spaces yield a Fock space. A function from one of them to itself need only be surjective and norm-preserving to be a linear unitary operator, by Wigner's theorem; such operators represent a symmetry. These spaces can be infinite-dimensional, have (*) an inner product, and are complete with respect to the norm induced by the inner product. They include the space of all square-integrable functions, big L 2. The observables of a quantum system, like spin and angular momentum, are represented by linear operators on them. For 10 points, identify these spaces named for the person who compiled a list of 23 unsolved problems in mathematics in 1900.
The GNS construction validates the use of these spaces because it derives them as representations of positive forms on a C-star algebra.
Hilbert spaces [prompt on "Banach (spaces)"]