View source for Rigged Hilbert space

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{{Technical|date=June 2020}}
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{{Short description|Construction linking the study of "bound" and continuous eigenvalues in functional analysis}}
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]]. They bring together the '[[bound state]]' ([[eigenvector]]) and '[[decomposition of spectrum (functional analysis)|continuous spectrum]]', in one place.

Using this notion, a version of the [[spectral theorem]] for [[unbounded operator]]s on Hilbert space can be formulated.<ref>{{eom|id=Rigged Hilbert space|first=R. A.|last=Minlos|authorlink=Robert Minlos}}</ref> "Rigged Hilbert spaces are well known as the structure which provides a proper mathematical meaning to the [[bra–ket notation|Dirac formulation of quantum mechanics]]."<ref>{{cite book |last1=Krasnoholovets |first1=Volodymyr |last2=Columbus |first2=Frank H. |title=New Research in Quantum Physics |date=2004 |publisher=Nova Science Publishers |isbn=978-1-59454-001-1 |page=79 |language=en}}</ref>

==Motivation==

A function such as
<math display="block"> x \mapsto e^{ix} , </math>
is an [[eigenfunction]] of the [[differential operator]]
<math display="block">-i\frac{d}{dx}</math>
on the [[real line]] {{math|'''R'''}}, but isn't [[square-integrable]] for the usual ([[Lebesgue measure|Lebesgue]]) measure on {{math|'''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 [[Distribution (mathematics)|distributions]], and a ''generalized eigenfunction'' theory was developed in the years after 1950.{{sfn|Gel'fand|Vilenkin|1964|pp=103-105}}

==Definition==

A '''rigged Hilbert space''' is a pair {{math|(''H'', Φ)}} with {{math|''H''}} a Hilbert space, {{math|Φ}} a dense subspace, such that {{math|Φ}} is given a [[topological vector space]] structure for which the [[inclusion map]]
<math display="block">i : \Phi \to H,</math>
is continuous.{{sfn|de la Madrid Modino|2001|pp=66-67}}{{sfn|van der Laan|2019|pp=21-22}} Identifying {{math|''H''}} with its dual space {{math|''H''<sup>*</sup>}}, the adjoint to {{math|''i''}} is the map
<math display="block">i^* : H = H^* \to \Phi^*.</math>

The duality pairing between {{math|Φ}} and {{math|Φ<sup>*</sup>}} is then compatible with the inner product on {{math|''H''}}, in the sense that:
<math display="block">\langle u, v\rangle_{\Phi\times\Phi^*} = (u, v)_H</math>
whenever <math>u \in \Phi\subset H</math> and <math>v \in H = H^* \subset \Phi^*</math>. In the case of complex Hilbert spaces, we use a Hermitian inner product; it will be complex linear in {{math|''u''}} (math convention) or {{math|''v''}} (physics convention), and conjugate-linear (complex anti-linear) in the other variable.

The triple <math> (\Phi,\,\,H,\,\,\Phi^*)</math> is often named the ''Gelfand triple'' (after [[Israel Gelfand]]). <math>H</math> is referred to as a pivot space.

Note that even though {{math|Φ}} is isomorphic to {{math|Φ<sup>*</sup>}} (via [[Riesz representation theorem|Riesz representation]]) if it happens that {{math|Φ}} is a Hilbert space in its own right, this isomorphism is ''not'' the same as the composition of the inclusion {{math|''i''}} with its adjoint {{math|''i''*}}
<math display="block">i^* i: \Phi\subset H = H^* \to \Phi^*.</math>

===Functional analysis approach===

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]] {{math|''H''}}, together with a subspace {{math|Φ}} which carries a [[finer topology]], that is one for which the natural inclusion
<math display="block"> \Phi \subseteq H </math>
is continuous.  It is [[Without loss of generality|no loss]] to assume that {{math|Φ}} is [[Dense set|dense]] in {{math|''H''}} for the Hilbert norm. We consider the inclusion of [[dual space]]s {{math|''H''<sup>*</sup>}} in {{math|Φ<sup>*</sup>}}. The latter, dual to {{math|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 {{math|Φ}} of type
<math display="block">\phi\mapsto\langle v,\phi\rangle</math>
for {{math|''v''}} in {{math|''H''}} are faithfully represented as distributions (because we assume {{math|Φ}} dense).

Now by applying the [[Riesz representation theorem]] we can identify {{math|''H''<sup>*</sup>}} with {{math|''H''}}. Therefore, the definition of ''rigged Hilbert space'' is in terms of a sandwich:
<math display="block">\Phi \subseteq H \subseteq \Phi^*. </math>

The most significant examples are those for which {{math|Φ}} is a [[nuclear space]]; this comment is an abstract expression of the idea that {{math|Φ}} consists of test functions and {{math|Φ*}} of the corresponding [[distribution (mathematics)|distributions]]. 

An example of a nuclear countably Hilbert space <math>\Phi</math> and its dual <math>\Phi^*</math> is the [[Schwartz space]] <math>\mathcal S(\mathbb R)</math> and the space of [[tempered distributions]] <math>\mathcal S'(\mathbb R)</math>, respectively, rigging the Hilbert space of [[square-integrable function]]s. As such, the rigged Hilbert space is given by{{sfn|de la Madrid Modino|2001|p=72}}
<math display="block">\mathcal{S}(\mathbb{R}) \subset L^2 (\mathbb{R}) \subset \mathcal{S}'(\mathbb{R}).</math>
Another example is given by [[Sobolev space]]s: Here (in the simplest case of Sobolev spaces on <math>\mathbb R^n</math>)
<math display="block">H = L^2(\mathbb R^n),\ \Phi = H^s(\mathbb R^n),\ \Phi^* = H^{-s}(\mathbb R^n),</math>
where <math>s > 0</math>.

==See also==
* [[Fourier inversion theorem]]
* [[Fourier_transform#Tempered_distributions|Fourier transform § Tempered distributions]]
* [[Self-adjoint_operator#Spectral_theorem|Self-adjoint operator § Spectral theorem]]

==Notes==
{{reflist}}

==References==
{{refbegin}}
* 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.)''
* [[Jean Dieudonné|J. Dieudonné]], ''Éléments d'analyse'' VII (1978). ''(See paragraphs 23.8 and 23.32)''
* {{cite book | last=Gel'fand | first=I. M. |author-link1=Israel_Gelfand| last2=Vilenkin | first2=N. Ya | author-link2=Naum_Yakovlevich_Vilenkin|title=Generalized Functions: Applications of Harmonic Analysis | publisher=Elsevier Science | publication-place=Burlington | date=1964 | isbn=978-1-4832-2974-4 |doi=10.1016/c2013-0-12221-0}}
* K. Maurin, ''Generalized Eigenfunction Expansions and Unitary Representations of Topological Groups'', Polish Scientific Publishers, Warsaw, 1968.
* {{cite thesis |last=de la Madrid Modino |first= R. |date=2001 |title= Quantum mechanics in rigged Hilbert space language|url=https://scholar.google.com/scholar?oi=bibs&cluster=2442809273695897641&btnI=1&hl=en |degree=  PhD |publisher= Universidad de Valladolid}}
* de la Madrid Modino, R. "The role of the rigged Hilbert space in Quantum Mechanics," Eur. J. Phys. 26, 287 (2005); [https://arxiv.org/abs/quant-ph/0502053 quant-ph/0502053].
* {{cite thesis |last=van der Laan |first=L. |date= July 2019 |title=Rigged Hilbert Space Theory for Hermitian and Quasi-Hermitian Observables |url=https://fse.studenttheses.ub.rug.nl/19933/ |degree=BSc |location=Groningen |publisher=Rijksuniversiteit Groningen |access-date=11 January 2025}}
{{refend}}

{{Functional analysis}}
{{Spectral theory}}
{{Hilbert space}}

[[Category:Generalized functions]]
[[Category:Hilbert spaces]]
[[Category:Spectral theory]]
[[Category:Schwartz distributions]]