Hilbert space: Difference between revisions
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In [[mathematics]], in particular in the branch known as [[functional analysis]], a '''Hilbert space''' is an [[inner product space]] that is [[completeness (mathematics)|complete]]. As such, it is automatically also a [[Banach space]]. Due to the presence of an inner product, a Hilbert space has additional useful geometric properties that are not found in a Banach space; these properties may be exploited to simplify analysis or to obtain stronger results. For example, in a Hilbert space there exist [[projection operator|projection operators]] onto closed subpaces of the Hilbert space. This makes Hilbert spaces extremely important in certain branches of mathematics such as [[optimization (mathematics)|optimization]] and [[approximation theory]]. As another example, any complex (real) continuous linear functional over a complex (real) Hilbert space can be expressed as the inner product of elements of that space with some fixed element of the space (this is known as the [[Riesz representation theorem]]). In mathematical terms, one says that a Hilbert space is [[isomorphism|isomorphic]] to its own [[dual space|dual]] and if the Hilbert space is real then it is actually its own dual. | In [[mathematics]], in particular in the branch known as [[functional analysis]], a '''Hilbert space''' is an [[inner product space]] that is [[completeness (mathematics)|complete]]. As such, it is automatically also a [[Banach space]]. Due to the presence of an inner product, a Hilbert space has additional useful geometric properties that are not found in a Banach space; these properties may be exploited to simplify analysis or to obtain stronger results. For example, in a Hilbert space there exist [[projection operator|projection operators]] onto closed subpaces of the Hilbert space. This makes Hilbert spaces extremely important in certain branches of mathematics such as [[optimization (mathematics)|optimization]] and [[approximation theory]]. As another example, any complex (real) continuous linear functional over a complex (real) Hilbert space can be expressed as the inner product of elements of that space with some fixed element of the space (this is known as the [[Riesz representation theorem]]). In mathematical terms, one says that a Hilbert space is [[isomorphism|isomorphic]] to its own [[dual space|dual]] and if the Hilbert space is real then it is actually its own dual. | ||
In [[physics]], Hilbert spaces play a fundamental role in the physical theory of [[quantum mechanics]]. The state (in the so-called [[ | In [[physics]], Hilbert spaces play a fundamental role in the physical theory of [[quantum mechanics]]. The state (in the so-called [[Schrödinger picture]]) of a quantum mechanical system is postulated to be a unit [[vector space|vector]] (i.e., a vector of norm 1) in some Hilbert space, and physical quantities or "observables" are postulated as self-adjoint operators on that Hilbert space. States serve to assign statistical properties to observables of the system. | ||
==See also== | ==See also== | ||
Revision as of 14:04, 13 October 2007
In mathematics, in particular in the branch known as functional analysis, a Hilbert space is an inner product space that is complete. As such, it is automatically also a Banach space. Due to the presence of an inner product, a Hilbert space has additional useful geometric properties that are not found in a Banach space; these properties may be exploited to simplify analysis or to obtain stronger results. For example, in a Hilbert space there exist projection operators onto closed subpaces of the Hilbert space. This makes Hilbert spaces extremely important in certain branches of mathematics such as optimization and approximation theory. As another example, any complex (real) continuous linear functional over a complex (real) Hilbert space can be expressed as the inner product of elements of that space with some fixed element of the space (this is known as the Riesz representation theorem). In mathematical terms, one says that a Hilbert space is isomorphic to its own dual and if the Hilbert space is real then it is actually its own dual.
In physics, Hilbert spaces play a fundamental role in the physical theory of quantum mechanics. The state (in the so-called Schrödinger picture) of a quantum mechanical system is postulated to be a unit vector (i.e., a vector of norm 1) in some Hilbert space, and physical quantities or "observables" are postulated as self-adjoint operators on that Hilbert space. States serve to assign statistical properties to observables of the system.