This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages)
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.
A function such as
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
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:
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 )
Formal definition (Gelfand triple)
Identifying H with its dual space H*, the adjoint to i is the map
The duality pairing between Φ and Φ* is then compatible with the inner product on H, in the sense that:
The triple 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*
- 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) , "Rigged_Hilbert_space", Encyclopedia of Mathematics, EMS Press