The observables of a quantum system, like spin and angular momentum, are represented by linear operators on them.
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)"]