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Motivation

A function such as x ↦ e i x , {\displaystyle x\mapsto e^{ix},} is an eigenfunction of the differential operator − i d d x {\displaystyle -i{\frac {d}{dx}}} on the real line R, but isn't square-integrable for the usual (Lebesgue) 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 distributions, and a generalized eigenfunction theory was developed in the years after 1950.³

Definition

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 : Φ → H , {\displaystyle i:\Phi \to H,} is continuous.⁴⁵ 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 Israel Gelfand). H {\displaystyle H} is referred to as a pivot space.

Note that even though Φ is isomorphic to Φ* (via Riesz representation) 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 ^{*}.}

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 H, together with a subspace Φ which carries a finer topology, that is one for which the natural inclusion Φ ⊆ H {\displaystyle \Phi \subseteq H} is continuous. It is no loss to assume that Φ is dense in H for the Hilbert norm. We consider the inclusion of dual spaces 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 linear functionals 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 Riesz representation theorem 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 nuclear space; this comment is an abstract expression of the idea that Φ consists of test functions and Φ* of the corresponding distributions.

An example of a nuclear countably Hilbert space Φ {\displaystyle \Phi } and its dual Φ ∗ {\displaystyle \Phi ^{*}} is the Schwartz space S ( R ) {\displaystyle {\mathcal {S}}(\mathbb {R} )} and the space of tempered distributions S ′ ( R ) {\displaystyle {\mathcal {S}}'(\mathbb {R} )} , respectively, rigging the Hilbert space of square-integrable functions. As such, the rigged Hilbert space is given by⁶ 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 Sobolev spaces: 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} .

See also

• Fourier inversion theorem
• Fourier transform § Tempered distributions
• Self-adjoint operator § Spectral theorem

Notes

• 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.)
• J. Dieudonné, Éléments d'analyse VII (1978). (See paragraphs 23.8 and 23.32)
• 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. {{cite book}}: ISBN / Date incompatibility (help)
• K. Maurin, Generalized Eigenfunction Expansions and Unitary Representations of Topological Groups, Polish Scientific Publishers, Warsaw, 1968.
• de la Madrid Modino, R. (2001). Quantum mechanics in rigged Hilbert space language (PhD thesis). Universidad de Valladolid.
• de la Madrid Modino, R. "The role of the rigged Hilbert space in Quantum Mechanics," Eur. J. Phys. 26, 287 (2005); quant-ph/0502053.
• 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.

References

1. Minlos, R. A. (2001) [1994], "Rigged Hilbert space", Encyclopedia of Mathematics, EMS Press /wiki/Robert_Minlos ↩

2. Krasnoholovets, Volodymyr; Columbus, Frank H. (2004). New Research in Quantum Physics. Nova Science Publishers. p. 79. ISBN 978-1-59454-001-1. 978-1-59454-001-1 ↩

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. https://doi.org/10.1016%2Fc2013-0-12221-0 ↩

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. https://scholar.google.com/scholar?oi=bibs&cluster=2442809273695897641&btnI=1&hl=en ↩

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. https://fse.studenttheses.ub.rug.nl/19933/ ↩

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. https://scholar.google.com/scholar?oi=bibs&cluster=2442809273695897641&btnI=1&hl=en ↩