Rigged Hilbert space

<h2 id="motivation">Motivation</h2>
A function such as

x
        ↦
        
          e
          
            i
            x
          
        
        ,
      
    
    {\displaystyle x\mapsto e^{ix},}

is an <a href="/facts/Eigenfunction/62EJS0XB">eigenfunction</a> of the <a href="/facts/Differential_operator/Nr15J0IE">differential operator</a>

−
        i
        
          
            d
            
              d
              x
            
          
        
      
    
    {\displaystyle -i{\frac {d}{dx}}}

on the <a href="/facts/Real_line/6eq42xX5">real line</a> R, but isn't <a href="/facts/Square-integrable/IASblcJJ">square-integrable</a> for the usual (<a href="/facts/Lebesgue_measure/8us9cGoW">Lebesgue</a>) measure on R. To properly consider this function as an eigenfunction requires some way of stepping outside the strict confines of the <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> theory. This was supplied by the apparatus of <a href="/facts/Distribution_(mathematics)/yfaJ4mUp">distributions</a>, and a generalized eigenfunction theory was developed in the years after 1950.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<h2 id="definition">Definition</h2>
A rigged Hilbert space is a pair (H, Φ) with H a Hilbert space, Φ a dense subspace, such that Φ is given a <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> structure for which the <a href="/facts/Inclusion_map/VUFNdaDH">inclusion map</a>

i
        :
        Φ
        →
        H
        ,
      
    
    {\displaystyle i:\Phi \to H,}

is continuous.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> Identifying H with its dual space H*, the adjoint to i is the map

i
          
            ∗
          
        
        :
        H
        =
        
          H
          
            ∗
          
        
        →
        
          Φ
          
            ∗
          
        
        .
      
    
    {\displaystyle i^{*}:H=H^{*}\to \Phi ^{*}.}

The duality pairing between Φ and Φ* is then compatible with the inner product on H, in the sense that:

⟨
        u
        ,
        v
        
          ⟩
          
            Φ
            ×
            
              Φ
              
                ∗
              
            
          
        
        =
        (
        u
        ,
        v
        
          )
          
            H
          
        
      
    
    {\displaystyle \langle u,v\rangle _{\Phi \times \Phi ^{*}}=(u,v)_{H}}

whenever 
 
 
 
 u
 ∈
 Φ
 ⊂
 H
 
 
 {\displaystyle u\in \Phi \subset H}
 
 and 
 
 
 
 v
 ∈
 H
 =
 
 H
 
 ∗
 
 
 ⊂
 
 Φ
 
 ∗
 
 
 
 
 {\displaystyle v\in H=H^{*}\subset \Phi ^{*}}
 
. In the case of complex Hilbert spaces, we use a Hermitian inner product; it will be complex linear in u (math convention) or v (physics convention), and conjugate-linear (complex anti-linear) in the other variable.
The triple 
 
 
 
 (
 Φ
 ,
 
 
 H
 ,
 
 
 
 Φ
 
 ∗
 
 
 )
 
 
 {\displaystyle (\Phi ,\,\,H,\,\,\Phi ^{*})}
 
 is often named the Gelfand triple (after <a href="/facts/Israel_Gelfand/TJeJouKs">Israel Gelfand</a>). 
 
 
 
 H
 
 
 {\displaystyle H}
 
 is referred to as a pivot space.
Note that even though Φ is isomorphic to Φ* (via <a href="/facts/Riesz_representation_theorem/9CjKDUII">Riesz representation</a>) if it happens that 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*

i
          
            ∗
          
        
        i
        :
        Φ
        ⊂
        H
        =
        
          H
          
            ∗
          
        
        →
        
          Φ
          
            ∗
          
        
        .
      
    
    {\displaystyle i^{*}i:\Phi \subset H=H^{*}\to \Phi ^{*}.}

<h3>Functional analysis approach</h3>
The concept of rigged Hilbert space places this idea in an abstract functional-analytic framework. Formally, a rigged Hilbert space consists of a <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> H, together with a subspace Φ which carries a <a href="/facts/Finer_topology/5Q9F2fjf">finer topology</a>, that is one for which the natural inclusion

Φ
        ⊆
        H
      
    
    {\displaystyle \Phi \subseteq H}

is continuous. It is <a href="/facts/Without_loss_of_generality/gVP4dmEe">no loss</a> to assume that Φ is <a href="/facts/Dense_set/nPT9Yxao">dense</a> in H for the Hilbert norm. We consider the inclusion of <a href="/facts/Dual_space/4Uw4Knz1">dual spaces</a> H* in Φ*. 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 <a href="/facts/Linear_functional/oGh7Asrz">linear functionals</a> on the subspace Φ of type

ϕ
        ↦
        ⟨
        v
        ,
        ϕ
        ⟩
      
    
    {\displaystyle \phi \mapsto \langle v,\phi \rangle }

for v in H are faithfully represented as distributions (because we assume Φ dense).
Now by applying the <a href="/facts/Riesz_representation_theorem/9CjKDUII">Riesz representation theorem</a> we can identify H* with H. Therefore, the definition of rigged Hilbert space is in terms of a sandwich:

Φ
        ⊆
        H
        ⊆
        
          Φ
          
            ∗
          
        
        .
      
    
    {\displaystyle \Phi \subseteq H\subseteq \Phi ^{*}.}

The most significant examples are those for which Φ is a <a href="/facts/Nuclear_space/4FLPWdfp">nuclear space</a>; this comment is an abstract expression of the idea that Φ consists of test functions and Φ* of the corresponding <a href="/facts/Distribution_(mathematics)/yfaJ4mUp">distributions</a>. 
An example of a nuclear countably Hilbert space 
 
 
 
 Φ
 
 
 {\displaystyle \Phi }
 
 and its dual 
 
 
 
 
 Φ
 
 ∗
 
 
 
 
 {\displaystyle \Phi ^{*}}
 
 is the <a href="/facts/Schwartz_space/B035faw7">Schwartz space</a> 
 
 
 
 
 
 S
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle {\mathcal {S}}(\mathbb {R} )}
 
 and the space of <a href="/facts/Tempered_distributions/yfaJ4mUp">tempered distributions</a> 
 
 
 
 
 
 
 S
 
 
 ′
 
 (
 
 R
 
 )
 
 
 {\displaystyle {\mathcal {S}}'(\mathbb {R} )}
 
, respectively, rigging the Hilbert space of <a href="/facts/Square-integrable_function/IASblcJJ">square-integrable functions</a>. As such, the rigged Hilbert space is given by<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

S
          
        
        (
        
          R
        
        )
        ⊂
        
          L
          
            2
          
        
        (
        
          R
        
        )
        ⊂
        
          
            
              S
            
          
          ′
        
        (
        
          R
        
        )
        .
      
    
    {\displaystyle {\mathcal {S}}(\mathbb {R} )\subset L^{2}(\mathbb {R} )\subset {\mathcal {S}}'(\mathbb {R} ).}

Another example is given by <a href="/facts/Sobolev_space/34qxWCyr">Sobolev spaces</a>: Here (in the simplest case of Sobolev spaces on 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
)

H
        =
        
          L
          
            2
          
        
        (
        
          
            R
          
          
            n
          
        
        )
        ,
         
        Φ
        =
        
          H
          
            s
          
        
        (
        
          
            R
          
          
            n
          
        
        )
        ,
         
        
          Φ
          
            ∗
          
        
        =
        
          H
          
            −
            s
          
        
        (
        
          
            R
          
          
            n
          
        
        )
        ,
      
    
    {\displaystyle H=L^{2}(\mathbb {R} ^{n}),\ \Phi =H^{s}(\mathbb {R} ^{n}),\ \Phi ^{*}=H^{-s}(\mathbb {R} ^{n}),}

where 
 
 
 
 s
 >
 0
 
 
 {\displaystyle s>0}
 
.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Fourier_inversion_theorem/aM1aMkf0">Fourier inversion theorem</a></li>
<li><a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform § Tempered distributions</a></li>
<li><a href="/facts/Self-adjoint_operator/VrH1nEAK">Self-adjoint operator § Spectral theorem</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>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, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-64305-2. (Provides a survey overview.)</li>
<li><a href="/facts/Jean_Dieudonn%25C3%25A9/gWagzMdu">J. Dieudonné</a>, Éléments d'analyse VII (1978). (See paragraphs 23.8 and 23.32)</li>
<li><a href="/facts/Israel_Gelfand/TJeJouKs">Gel'fand, I. M.</a>; <a href="/facts/Naum_Yakovlevich_Vilenkin/XxLLu07U">Vilenkin, N. Ya</a> (1964). Generalized Functions: Applications of Harmonic Analysis. Burlington: Elsevier Science. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fc2013-0-12221-0">10.1016/c2013-0-12221-0</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4832-2974-4. {{cite book}}: ISBN / Date incompatibility (help)</li>
<li>K. Maurin, Generalized Eigenfunction Expansions and Unitary Representations of Topological Groups, Polish Scientific Publishers, Warsaw, 1968.</li>
<li>de la Madrid Modino, R. (2001). <a href="https://scholar.google.com/scholar?oi=bibs&cluster=2442809273695897641&btnI=1&hl=en">Quantum mechanics in rigged Hilbert space language</a> (PhD thesis). Universidad de Valladolid.</li>
<li>de la Madrid Modino, R. "The role of the rigged Hilbert space in Quantum Mechanics," Eur. J. Phys. 26, 287 (2005); <a href="https://arxiv.org/abs/quant-ph/0502053">quant-ph/0502053</a>.</li>
<li>van der Laan, L. (July 2019). <a href="https://fse.studenttheses.ub.rug.nl/19933/">Rigged Hilbert Space Theory for Hermitian and Quasi-Hermitian Observables</a> (BSc thesis). Groningen: Rijksuniversiteit Groningen. Retrieved 11 January 2025.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Minlos, R. A. (2001) [1994], "Rigged Hilbert space", Encyclopedia of Mathematics, EMS Press <a href="/wiki/Robert_Minlos" target="_blank">/wiki/Robert_Minlos</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Krasnoholovets, Volodymyr; Columbus, Frank H. (2004). New Research in Quantum Physics. Nova Science Publishers. p. 79. ISBN 978-1-59454-001-1. <a href="978-1-59454-001-1" target="_blank">978-1-59454-001-1</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Gel'fand & Vilenkin 1964, pp. 103–105. - Gel'fand, I. M.; Vilenkin, N. Ya (1964). Generalized Functions: Applications of Harmonic Analysis. Burlington: Elsevier Science. doi:10.1016/c2013-0-12221-0. ISBN 978-1-4832-2974-4. <a href="https://doi.org/10.1016%2Fc2013-0-12221-0" target="_blank">https://doi.org/10.1016%2Fc2013-0-12221-0</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">de la Madrid Modino 2001, pp. 66–67. - de la Madrid Modino, R. (2001). Quantum mechanics in rigged Hilbert space language (PhD thesis). Universidad de Valladolid. <a href="https://scholar.google.com/scholar?oi=bibs&cluster=2442809273695897641&btnI=1&hl=en" target="_blank">https://scholar.google.com/scholar?oi=bibs&cluster=2442809273695897641&btnI=1&hl=en</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">van der Laan 2019, pp. 21–22. - van der Laan, L. (July 2019). Rigged Hilbert Space Theory for Hermitian and Quasi-Hermitian Observables (BSc thesis). Groningen: Rijksuniversiteit Groningen. Retrieved 11 January 2025. <a href="https://fse.studenttheses.ub.rug.nl/19933/" target="_blank">https://fse.studenttheses.ub.rug.nl/19933/</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">de la Madrid Modino 2001, p. 72. - de la Madrid Modino, R. (2001). Quantum mechanics in rigged Hilbert space language (PhD thesis). Universidad de Valladolid. <a href="https://scholar.google.com/scholar?oi=bibs&cluster=2442809273695897641&btnI=1&hl=en" target="_blank">https://scholar.google.com/scholar?oi=bibs&cluster=2442809273695897641&btnI=1&hl=en</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
</ol>

Rigged Hilbert space open-in-new

Rigged Hilbert space