Rigged Hilbert space

Last updated April 28, 2024

In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study spectral theory in the broad sense.^{[ vague ]} They bring together the 'bound state' (eigenvector) and 'continuous spectrum', in one place.

Motivation

A function such as

x\mapsto e^{ix},

is an eigenfunction of the differential operator

-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.

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

\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

\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:

\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. Also, a simple example is given by Sobolev spaces: Here (in the simplest case of Sobolev spaces on $\mathbb {R} ^{n}$ )

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

where $s>0$ .

Formal definition (Gelfand triple)

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.

Identifying $H$ with its dual space $H *$ , the adjoint to $i$ is the map

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:

\langle u,v\rangle _{\Phi \times \Phi ^{*}}=(u,v)_{H}

whenever $u\in \Phi \subset H$ and $v\in H=H^{*}\subset \Phi ^{*}$ . 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 $(\Phi ,\,\,H,\,\,\Phi ^{*})$ is often named the "Gelfand triple" (after the mathematician Israel Gelfand).

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:\Phi \subset H=H^{*}\to \Phi ^{*}.

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References

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)
I. M. Gelfand and N. Ya. Vilenkin. Generalized Functions, vol. 4: Some Applications of Harmonic Analysis. Rigged Hilbert Spaces. Academic Press, New York, 1964.
K. Maurin, Generalized Eigenfunction Expansions and Unitary Representations of Topological Groups, Polish Scientific Publishers, Warsaw, 1968.
R. de la Madrid, "Quantum Mechanics in Rigged Hilbert Space Language," PhD Thesis (2001).
R. de la Madrid, "The role of the rigged Hilbert space in Quantum Mechanics," Eur. J. Phys. 26, 287 (2005); quant-ph/0502053.
Minlos, R.A. (2001) [1994], "Rigged_Hilbert_space", Encyclopedia of Mathematics , EMS Press

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v t e Spectral theory and ^*-algebras
Basic concepts	Involution/-algebra Banach algebra B-algebra C-algebra Noncommutative topology Projection-valued measure Spectrum Spectrum of a C-algebra Spectral radius Operator space
Main results	Gelfand–Mazur theorem Gelfand–Naimark theorem Gelfand representation Polar decomposition Singular value decomposition Spectral theorem Spectral theory of normal C*-algebras
Special Elements/Operators	Isospectral Normal operator Hermitian/Self-adjoint operator Unitary operator Unit
Spectrum	Krein–Rutman theorem Normal eigenvalue Spectrum of a C*-algebra Spectral radius Spectral asymmetry Spectral gap
Decomposition	Decomposition of a spectrum Continuous Point Residual Approximate point Compression Direct integral Discrete Spectral abscissa
Spectral Theorem	Borel functional calculus Min-max theorem Positive operator-valued measure Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
Special algebras	Amenable Banach algebra With an Approximate identity Banach function algebra Disk algebra Nuclear C*-algebra Uniform algebra Von Neumann algebra Tomita–Takesaki theory
Finite-Dimensional	Alon–Boppana bound Bauer–Fike theorem Numerical range Schur–Horn theorem
Generalizations	Dirac spectrum Essential spectrum Pseudospectrum Structure space (Shilov boundary)
Miscellaneous	Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Fredholm theory Limiting absorption principle Schröder–Bernstein theorems for operator algebras Sherman–Takeda theorem Unbounded operator
Examples	Wiener algebra
Applications	Almost Mathieu operator Corona theorem Hearing the shape of a drum (Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law Wiener–Khinchin theorem

v t e Hilbert spaces
Basic concepts	Adjoint Inner product and L-semi-inner product Hilbert space and Prehilbert space Orthogonal complement Orthonormal basis
Main results	Bessel's inequality Cauchy–Schwarz inequality Riesz representation
Other results	Hilbert projection theorem Parseval's identity Polarization identity (Parallelogram law)
Maps	Compact operator on Hilbert space Densely defined Hermitian form Hilbert–Schmidt Normal Self-adjoint Sesquilinear form Trace class Unitary
Examples	Cⁿ(K) with K compact & n<∞ Segal–Bargmann F