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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.
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 v {\displaystyle v} is a unit vector of H {\displaystyle \mathbb {H} } then ω ( A ) = v , A v {\displaystyle \omega (A)=\langle v,Av\rangle } is a state on the C*-algebra, meaning the ...
The name spectral theory was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many variables. The original spectral theorem was therefore conceived as a version of the theorem on principal axes of an ellipsoid , in an infinite-dimensional setting.
One key observation about this picture is that L 2,h (D) may be identified with the space of holomorphic (n,0)-forms on D, via multiplication by . Since the L 2 {\displaystyle L^{2}} inner product on this space is manifestly invariant under biholomorphisms of D, the Bergman kernel and the associated Bergman metric are therefore automatically ...
In mathematics, a Hilbert manifold is a manifold modeled on Hilbert spaces. Thus it is a separable Hausdorff space in which each point has a neighbourhood homeomorphic to an infinite dimensional Hilbert space. The concept of a Hilbert manifold provides a possibility of extending the theory of manifolds to infinite-dimensional setting.
The term Hilbert geometry may refer to several things named after David Hilbert: . Hilbert's axioms, a modern axiomatization of Euclidean geometry; Hilbert space, a space in many ways resembling a Euclidean space, but in important instances infinite-dimensional
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If is a locally compact Hausdorff space and a vector bundle over with projection : a Hermitian metric, then the space of continuous sections of is a Hilbert ()-module. Given sections σ , ρ {\displaystyle \sigma ,\rho } of E {\displaystyle E} and f ∈ C ( X ) {\displaystyle f\in C(X)} the right action is defined by