In
mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the
distribution Distribution may refer to:
Mathematics
*Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
*Probability distribution, the probability of a particular value or value range of a varia ...
and
square-integrable
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value ...
aspects of
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 (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
. Such spaces were introduced to study
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 ...
in the broad sense. They bring together the '
bound state
Bound or bounds may refer to:
Mathematics
* Bound variable
* Upper and lower bounds, observed limits of mathematical functions
Physics
* Bound state, a particle that has a tendency to remain localized in one or more regions of space
Geography
* ...
' (
eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
) and '
continuous spectrum
In physics, a continuous spectrum usually means a set of attainable values for some physical quantity (such as energy or wavelength) that is best described as an interval of real numbers, as opposed to a discrete spectrum, a set of attain ...
', in one place.
Motivation
A function such as the canonical
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sa ...
of the real line into the complex plane
:
is an
eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, thi ...
of the
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 that accepts a function and return ...
:
on the
real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
R, but isn't
square-integrable
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value ...
for the usual
Borel measure
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below.
...
on R. To properly consider this function as an eigenfunction requires some way of stepping outside the strict confines of the
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
theory. This was supplied by the apparatus of
Schwartz distribution
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...
s, 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
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
''H'', together with a subspace Φ which carries a
finer topology
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies.
Definition
A topology on a set may be defined as the ...
, 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
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
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
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the s ...
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
:''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.''
The Riesz representation theorem, sometimes called th ...
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
In mathematics, nuclear spaces are topological vector spaces that can be viewed as a generalization of finite dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite different from Hilbert spaces, ...
; 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 space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
s: Here (in the simplest case of Sobolev spaces on
)
:
where
.
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
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
structure for which the
inclusion map
In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B:
\iota : A\rightarrow B, \qquad \iota ...
''i'' is continuous.
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:
:
whenever
and
. 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
is often named the "Gelfand triple" (after the mathematician
Israel Gelfand
Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand ( yi, ישראל געלפֿאַנד, russian: Изра́иль Моисе́евич Гельфа́нд, uk, Ізраїль Мойсейович Гел� ...
).
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''*
:
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, . ''(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
*
{{SpectralTheory
Hilbert space
Spectral theory
Generalized functions