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| In [[mathematics]], a '''rigged Hilbert space''' ('''Gelfand triple''', '''nested Hilbert space''', '''equipped Hilbert space''') is a construction designed to link the [[distribution (mathematics)|distribution]] and [[square-integrable]] aspects of [[functional analysis]]. Such spaces were introduced to study [[spectral theory]] in the broad sense.{{vague|It does something that isn't everything|date=January 2012}} They can bring together the '[[bound state]]' ([[eigenvector]]) and '[[Decomposition of spectrum (functional analysis)|continuous spectrum]]', in one place.
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| ==Motivation==
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| A function such as the canonical homomorphism of the real line into the complex plane
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| :<math> x \mapsto e^{ix} , </math>
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| is an [[eigenvector]] of the [[differential operator]]
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| :<math>-i\frac{d}{dx}</math>
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| on the [[real line]] '''R''', but isn't [[square-integrable]] for the usual [[Borel measure]] on '''R'''. To properly consider this function as an eigenfunction requires some way of stepping outside the strict confines of the [[Hilbert space]] theory. This was supplied by the apparatus of [[Schwartz distribution]]s, and a ''generalized eigenfunction'' theory was developed in the years after 1950.
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| ==Functional analysis approach==
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| The concept of rigged Hilbert space places this idea in an abstract functional-analytic framework. Formally, a rigged Hilbert space consists of a [[Hilbert space]] ''H'', together with a subspace Φ which carries a [[finer topology]], that is one for which the natural inclusion
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| :<math> \Phi \subseteq H </math>
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| is continuous. It is [[Without loss of generality|no loss]] to assume that Φ is [[Dense_set|dense]] in ''H'' for the Hilbert norm. We consider the inclusion of [[dual space]]s ''H''<sup>*</sup> in Φ<sup>*</sup>. The latter, dual to Φ in its 'test function' topology, is realised as a space of distributions or generalised functions of some sort, and the [[linear functional]]s on the subspace Φ of type
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| :<math>\phi\mapsto\langle v,\phi\rangle</math> | |
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| for ''v'' in ''H'' are faithfully represented as distributions (because we assume Φ dense).
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| Now by applying the [[Riesz representation theorem]] we can identify ''H''<sup>*</sup> with ''H''. Therefore the definition of ''rigged Hilbert space'' is in terms of a sandwich:
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| :<math>\Phi \subseteq H \subseteq \Phi^*. </math>
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| The most significant examples are those for which Φ is a [[nuclear space]]; this comment is an abstract expression of the idea that Φ consists of test functions and Φ* of the corresponding [[distribution (mathematics)|distributions]].
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| ==Formal definition (Gelfand triple)==
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| A '''rigged Hilbert space''' is a pair (''H'',Φ) with ''H'' a Hilbert space, Φ a dense subspace, such that Φ is given a [[topological vector space]] structure for which the [[inclusion map]] ''i'' is continuous.
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| Identifying ''H'' with its dual space ''H<sup>*</sup>'', the adjoint to ''i'' is the map
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| :<math>i^*:H=H^*\to\Phi^*.</math>
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| The duality pairing between Φ and Φ<sup>*</sup> has to be compatible with the inner product on ''H'', in the sense that:
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| :<math>\langle u, v\rangle_{\Phi\times\Phi^*} = (u, v)_H</math>
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| whenever <math>u\in\Phi\subset H</math> and <math>v \in H=H^* \subset \Phi^*</math>.
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| The specific triple <math> (\Phi,\,\,H,\,\,\Phi^*)</math> is often named the "Gelfand triple" (after the mathematician [[Israel Gelfand]]).
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| Note that even though Φ is isomorphic to Φ<sup>*</sup> if Φ is a Hilbert space in its own right, this isomorphism is ''not'' the same as the composition of the inclusion ''i'' with its adjoint ''i''*
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| :<math>i^* i:\Phi\subset H=H^*\to\Phi^*.</math>
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| ==References==
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| * J.-P. Antoine, ''Quantum Mechanics Beyond Hilbert Space'' (1996), appearing in ''Irreversibility and Causality, Semigroups and Rigged Hilbert Spaces'', Arno Bohm, Heinz-Dietrich Doebner, Piotr Kielanowski, eds., Springer-Verlag, ISBN 3-540-64305-2. ''(Provides a survey overview.)''
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| * [[Jean Dieudonné]], ''Éléments d'analyse'' VII (1978). ''(See paragraphs 23.8 and 23.32)''
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| * [[Israel Gelfand|I. M. Gelfand]] and N. J. Vilenkin. Generalized Functions, vol. 4: Some Applications of Harmonic Analysis. Rigged Hilbert Spaces. Academic Press, New York, 1964.
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| * R. de la Madrid, "The role of the rigged Hilbert space in Quantum Mechanics," Eur. J. Phys. 26, 287 (2005); [http://arxiv.org/abs/quant-ph/0502053 quant-ph/0502053].
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| * K. Maurin, ''Generalized Eigenfunction Expansions and Unitary Representations of Topological Groups'', Polish Scientific Publishers, Warsaw, 1968.
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| *{{eom|id=Rigged_Hilbert_space|first=R.A.|last= Minlos}}
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| [[Category:Hilbert space]]
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| [[Category:Spectral theory]]
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| [[Category:Generalized functions]]
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