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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.
Canonically cited as Dunford and Schwartz, [1] the textbook has been referred to as "the definitive work" on linear operators. [2]: 2 The work began as a written set of solutions to the problems for Dunford's graduate course in linear operators at Yale. [3]: 30 [1] Schwartz, a prodigy, had taken his undergraduate degree at Yale in 1948, age 18 ...
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.
The predual of a von Neumann algebra is in fact unique up to isomorphism. Some authors use "von Neumann algebra" for the algebras together with a Hilbert space action, and "W*-algebra" for the abstract concept, so a von Neumann algebra is a W*-algebra together with a Hilbert space and a suitable faithful unital action on the Hilbert space.
The Hilbert Curve is commonly used among rendering images or videos. Common programs such as Blender and Cinema 4D use the Hilbert Curve to trace the objects, and render the scene. [citation needed] The slicer software used to convert 3D models into toolpaths for a 3D printer typically has the Hilbert curve as an option for an infill pattern.
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 Hilbert metric , a metric that makes a bounded convex subset of a Euclidean space into an unbounded metric space