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Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space. A Hilbert space is a special case of a Banach space. Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz.
This definition applies to a Banach space, but of course other types of space exist as well; for example, topological vector spaces include Banach spaces, but can be more general. [12] [13] On the other hand, Banach spaces include Hilbert spaces, and it is these spaces that find the greatest application and the richest theoretical results. [14]
We say that H is a reproducing kernel Hilbert space if, for all in , is ... “An introduction to the theory of reproducing kernel Hilbert spaces,” https: ...
The algebraic tensor product of two Hilbert spaces A and B has a natural positive definite sesquilinear form (scalar product) induced by the sesquilinear forms of A and B.So in particular it has a natural positive definite quadratic form, and the corresponding completion is a Hilbert space A ⊗ B, called the (Hilbert space) tensor product of A and B.
As such, quantum states form a ray in projective Hilbert space, not a vector. Many textbooks fail to make this distinction, which could be partly a result of the fact that the Schrödinger equation itself involves Hilbert-space "vectors", with the result that the imprecise use of "state vector" rather than ray is very difficult to avoid. [5]
If the C*-algebra is the algebra of all bounded operators on a Hilbert space , then the bounded observables are just the bounded self-adjoint operators on .If is a unit vector of then = , is a state on the C*-algebra, meaning the unit vectors (up to scalar multiplication) give the states of the system.
In the Hilbert space formulation of quantum mechanics as presented by von Neumann, a physical observable is represented by some (possibly unbounded) densely defined self-adjoint operator A on a Hilbert space H. A has a spectral decomposition, which is a projection-valued measure E defined on the Borel subsets of R.
The term C*-algebra was introduced by I. E. Segal in 1947 to describe norm-closed subalgebras of B(H), namely, the space of bounded operators on some Hilbert space H. 'C' stood for 'closed'. [2] [3] In his paper Segal defines a C*-algebra as a "uniformly closed, self-adjoint algebra of bounded operators on a Hilbert space". [4]