Rigged Hilbert space
Encyclopedia
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. They can bring together the 'bound state
' (eigenvector) and 'continuous spectrum
', in one place.
,
which is in an obvious sense an eigenvector of the differential operator
on the real line
R, is not square-integrable for the usual Borel measure on R, this requires some way of stepping outside the strict confines of the Hilbert space
theory. This was supplied by the apparatus of Schwartz distributions, and a generalized eigenfunction theory was developed in the years after 1950.
H, together with a subspace Φ which carries a finer topology, that is one for which the natural inclusion
is continuous. It is no loss to assume that Φ is dense in H for the Hilbert norm. We consider the inclusion of dual space
s 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 functional
s on the subspace Φ of type
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:
The most significant examples are 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
.
structure for which the inclusion map
i is continuous.
Identifying H with its dual space H*, the adjoint to i is the map
.
The duality pairing between Φ and Φ* has to be compatible with the inner product on H, in the sense that:
whenever
and 
.
The specific triple
is often named the "Gelfand triple" (after the mathematician Israel Gelfand
).
Note that even though Φ is isomorphic to Φ* if 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*
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
and square-integrable aspects of functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
. Such spaces were introduced to study spectral theory
Spectral theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of...
in the broad sense. They can bring together the 'bound state
Bound state
In physics, a bound state describes a system where a particle is subject to a potential such that the particle has a tendency to remain localised in one or more regions of space...
' (eigenvector) and 'continuous spectrum
Decomposition of spectrum (functional analysis)
In mathematics, especially functional analysis, the spectrum of an operator generalizes the notion of eigenvalues. Given an operator, it is sometimes useful to break up the spectrum into various parts...
', in one place.
Motivation
Since a function such as the canonical homomorphism of the real line into the complex plane,which is in an obvious sense an eigenvector of the differential operator
Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
on the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
R, is not square-integrable for the usual Borel measure on R, this requires some way of stepping outside the strict confines of the Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
theory. This was supplied by the apparatus of Schwartz 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 abstract functional-analytic framework. Formally, a rigged Hilbert space consists of a Hilbert spaceHilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
H, together with a subspace Φ which carries a finer topology, that is one for which the natural inclusion
is continuous. It is no loss to assume that Φ is dense in H for the Hilbert norm. We consider the inclusion of dual space
Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
s 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 functional
Linear functional
In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars. In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...
s on the subspace Φ of type
for v in H are faithfully represented as distributions (because we assume Φ dense).
Now by applying the Riesz representation theorem
Riesz representation theorem
There are several well-known theorems in functional analysis known as the Riesz representation theorem. They are named in honour of Frigyes Riesz.- The Hilbert space 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 for which Φ is a nuclear space
Nuclear space
In mathematics, a nuclear space is a topological vector space with many of the good properties of finite-dimensional vector spaces. The topology on them can be defined by a family of seminorms whose unit balls decrease rapidly in size...
; this comment is an abstract expression of the idea that Φ consists of test functions and Φ* of the corresponding distributions
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
.
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 spaceTopological vector space
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
structure for which the inclusion map
Inclusion map
In mathematics, if A is a subset of B, then the inclusion map is the function i that sends each element, x of A to x, treated as an element of B:i: A\rightarrow B, \qquad i=x....
i is continuous.
Identifying H with its dual space H*, the adjoint to i is the map
.The duality pairing between Φ and Φ* has to be compatible with the inner product on H, in the sense that:
whenever
and .The specific triple
is often named the "Gelfand triple" (after the mathematician Israel GelfandIsrael Gelfand
Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand was a Soviet mathematician who made major contributions to many branches of mathematics, including group theory, representation theory and functional analysis...
).
Note that even though Φ is isomorphic to Φ* if 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*