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The exact nature of this Hilbert space is dependent on the system; for example, the position and momentum states for a single non-relativistic spin zero particle is the space of all square-integrable functions, while the states for the spin of a single proton are unit elements of the two-dimensional complex Hilbert space of spinors.
Consider a separable Hilbert space (H, , ) equipped with a Gaussian measure γ. As stated in the article on the Vitali covering theorem , the Vitali covering theorem fails for Gaussian measures on infinite-dimensional Hilbert spaces.
The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems. For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.
Hilbert–Schmidt integral operators are both continuous and compact. [3] The concept of a Hilbert–Schmidt operator may be extended to any locally compact Hausdorff spaces. Specifically, let L 2 (X) be a separable Hilbert space and X a locally compact Hausdorff space equipped with a positive Borel measure.
The Hilbert–Schmidt operators form a two-sided *-ideal in the Banach algebra of bounded operators on H. They also form a Hilbert space, denoted by B HS (H) or B 2 (H), which can be shown to be naturally isometrically isomorphic to the tensor product of Hilbert spaces, where H ∗ is the dual space of H.
Every real Hilbert space can be extended to be a dense subset of a unique (up to bijective isometry) complex Hilbert space, called its complexification, which is why Hilbert spaces are often automatically assumed to be complex. Real and complex Hilbert spaces have in common many, but by no means all, properties and results/theorems.
The simplest example of a direct integral are the L 2 spaces associated to a (σ-finite) countably additive measure μ on a measurable space X.Somewhat more generally one can consider a separable Hilbert space H and the space of square-integrable H-valued functions
The Hilbert space tensor product of two Hilbert spaces is the completion of their algebraic tensor product. One can define a tensor product of von Neumann algebras (a completion of the algebraic tensor product of the algebras considered as rings), which is again a von Neumann algebra, and act on the tensor product of the corresponding Hilbert ...