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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 ...
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Hilbert spaces can be completely classified: there is a unique Hilbert space up to isomorphism for every cardinality of the orthonormal basis. [5] Finite-dimensional Hilbert spaces are fully understood in linear algebra , and infinite-dimensional separable Hilbert spaces are isomorphic to ℓ 2 ( ℵ 0 ) {\displaystyle \ell ^{\,2}(\aleph _{0})\,} .
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]
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.
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
In quantum field theory, it is expected that the Hilbert space is also the space on the configuration space of the field, which is infinite dimensional, with respect to some Borel measure naturally defined. However, it is often hard to define a concrete Borel measure on the classical configuration space, since the integral theory on infinite ...