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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.
The time evolution of the state is given by a differentiable function from the real numbers R, representing instants of time, to the Hilbert space of system states. This map is characterized by a differential equation as follows: If |ψ(t) denotes the state of the system at any one time t, the following Schrödinger equation holds:
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 version for Hilbert spaces can for example be found in (Schwartz 1969, p. 21). [2] If H 1 is a separable space (in particular, if it is a Euclidean space) the result is true in Zermelo–Fraenkel set theory; for the fully general case, it appears to need some form of the axiom of choice; the Boolean prime ideal theorem is known to be ...
In mathematical analysis, the Hilbert–Schmidt theorem, also known as the eigenfunction expansion theorem, is a fundamental result concerning compact, self-adjoint operators on Hilbert spaces. In the theory of partial differential equations , it is very useful in solving elliptic boundary value problems .
Let be a Hilbert space defined over the real numbers, assumed to be infinite dimensional and separable. A cylinder set in H {\displaystyle H} is a set defined in terms of the values of a finite collection of linear functionals on H {\displaystyle H} .
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
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.