Search results
Results From The WOW.Com Content Network
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.
Every real Hilbert space can be extended to be a dense subset of a unique (up to bijective isometry) complex Hilbert space, called its complexification, which is why Hilbert spaces are often automatically assumed to be complex. Real and complex Hilbert spaces have in common many, but by no means all, properties and results/theorems.
The space ([,]) of continuous real-valued functions on the unit interval [,] with the metric of uniform convergence is a separable space, since it follows from the Weierstrass approximation theorem that the set [] of polynomials in one variable with rational coefficients is a countable dense subset of ([,]).
Two results of David Preiss (1981 and 1983) show the kind of difficulties that one can expect to encounter in this setting: There is a Gaussian measure γ on a separable Hilbert space H and a Borel set M ⊆ H so that, for γ -almost all x ∈ H , lim r → 0 γ ( M ∩ B r ( x ) ) γ ( B r ( x ) ) = 1. {\displaystyle \lim _{r\to 0}{\frac ...
In the mathematical discipline of functional analysis, the concept of a compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators (representable by finite-dimensional matrices) in the topology induced by the operator norm.
The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems. For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.
The first two definitions describe a von Neumann algebra concretely as a set of operators acting on some given Hilbert space. Sakai (1971) showed that von Neumann algebras can also be defined abstractly as C*-algebras that have a predual ; in other words the von Neumann algebra, considered as a Banach space , is the dual of some other Banach ...
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.