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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 ...
Geometry and the Imagination is the English translation of the 1932 book Anschauliche Geometrie by David Hilbert and Stefan Cohn-Vossen. [1] The book was based on a series of lectures Hilbert made in the winter of 1920–21. The book is an attempt to present some then-current mathematical thought to "contribute to a more just appreciation of ...
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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.
For a two-dimensional Hilbert space, the space of all such states is the complex projective line. This is the Bloch sphere, which can be mapped to the Riemann sphere . The Bloch sphere is a unit 2-sphere , with antipodal points corresponding to a pair of mutually orthogonal state vectors.
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Removing five axioms mentioning "plane" in an essential way, namely I.4–8, and modifying III.4 and IV.1 to omit mention of planes, yields an axiomatization of Euclidean plane geometry. Hilbert's axioms, unlike Tarski's axioms, do not constitute a first-order theory because the axioms V.1–2 cannot be expressed in first-order logic.