What is a Hilbert space in algebra?
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. It's essentially just an infinitely dimensional vector space. ﻿A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.
Hilbert space
Two vectors and in a Hilbert space are orthogonal when . The notation for this is . More generally, when is a subset in , the notation means that is orthogonal to every element from .
Hilbert scheme
In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomials. The basic theory of Hilbert schemes was developed by . Hironaka's example shows that non-projective varieties need not have Hilbert schemes.
Vector space
Imposing boundedness conditions not only on the function, but also on its derivatives leads to Sobolev spaces.
Complete inner product spaces are known as "Hilbert spaces", in honor of David Hilbert.
The Hilbert space "L"(Ω), with inner product given by
where formula_53 denotes the complex conjugate of "g"("x"), is a key case.
Hilbert space
Hilbert spaces allow for many elliptic partial differential equations to be formulated in a similar way, and the Lax–Milgram theorem is then a basic tool in their analysis. With suitable modifications, similar techniques can be applied to parabolic partial differential equations and certain hyperbolic partial differential equations.
Reproducing kernel Hilbert space
and
Calkin algebra
In functional analysis, the Calkin algebra, named after John Wilson Calkin, is the quotient of "B"("H"), the ring of bounded linear operators on a separable infinite-dimensional Hilbert space "H", by the ideal "K"("H") of compact operators. Here the addition in "B"("H") is addition of operators and the multiplication in "B"("H") is composition of operators; it is easy to verify that these operations make "B"("H") into a ring. When scalar multiplication is also included, "B"("H") becomes in fact an algebra over the same field over which "H" is a Hilbert space.
Hilbert space
A Hilbert space is separable if and only if it admits a countable orthonormal basis. All infinite-dimensional separable Hilbert spaces are therefore isometrically isomorphic to .
CCR and CAR algebras
Let formula_16 be a Hilbert space. In the theory of operator algebras the CAR algebra is the unique C*-completion of the complex unital *-algebra generated by elements formula_37 subject to the relations
for any formula_28, formula_43.
When formula_16 is separable the CAR algebra is an AF algebra and in the special case formula_16 is infinite dimensional it is often written as formula_46.
Hilbert space
One of the most familiar examples of a Hilbert space is the Euclidean space consisting of three-dimensional vectors, denoted by , and equipped with the dot product. The dot product takes two vectors and , and produces a real number . If and are represented in Cartesian coordinates, then the dot product is defined by
Sobolev space
The Sobolev space formula_96 is also denoted by formula_97 It is a Hilbert space, with an important subspace formula_98 defined to be the closure of the infinitely differentiable functions compactly supported in formula_1 in formula_97 The Sobolev norm defined above reduces here to
Spectrum (functional analysis)
For example, consider the Hilbert space formula_24, that consists of all bi-infinite sequences of real numbers
that have a finite sum of squares formula_26. The bilateral shift operator formula_2 simply displaces every element of the sequence by one position; namely if formula_28 then formula_29 for every integer formula_30. The eigenvalue equation formula_31 has no solution in this space, since it implies that all the values formula_32 have the same absolute value (if formula_33) or are a geometric progression (if formula_34); either way, the sum of their squares would not be finite. However, the operator formula_20 is not invertible if formula_36. For example, the sequence formula_37 such that formula_38 is in formula_24; but there is no sequence formula_40 in formula_24 such that formula_42 (that is, formula_43 for all formula_30).
Basil Hiley
In 2000, Brown and Hiley showed that the Schrödinger equation can be written in a purely algebraic form that is independent of any representation in a Hilbert space. This algebraic description is formulated in terms of two operator equations. The first of these (formulated in terms of the commutator) represents an alternative form of the quantum Liouville equation, which is well known to describe the conservation of probability, the second (formulated in terms of the anticommutator), which they dubbed the "quantum phase equation", describes the conservation of energy. This algebraic description in turn gives rise to descriptions in terms of multiple vector spaces, which Brown and Hiley call "shadow phase spaces" (adopting the term "shadow" from Michał Heller). These shadow phase space descriptions include the descriptions in terms of the "x"-space of the Bohm trajectory description, of the quantum phase space, and of the "p"-space. In the classical limit, the shadow phase spaces converge to one unique phase space. In their algebraic formulation of quantum mechanics the equation of motion takes on the same form as in the Heisenberg picture, except that the "bra" and "ket" in the bra–ket notation each stand for an element of the algebra and that the Heisenberg time evolution is an inner automorphism in the algebra.
Hilbert space
Many of the applications of Hilbert spaces exploit the fact that Hilbert spaces support generalizations of simple geometric concepts like projection and change of basis from their usual finite dimensional setting. In particular, the spectral theory of continuous self-adjoint linear operators on a Hilbert space generalizes the usual spectral decomposition of a matrix, and this often plays a major role in applications of the theory to other areas of mathematics and physics.
Unitary transformation
where formula_2 and formula_3 are Hilbert spaces, such that
Abelian von Neumann algebra
for measurable families of Hilbert spaces {"H"}, {"K"}. If "U" : "H" → "K" is a unitary such that
Projection (linear algebra)
When the vector space formula_38 has an inner product and is complete (is a Hilbert space) the concept of orthogonality can be used. An orthogonal projection is a projection for which the range formula_41 and the null space formula_4 are orthogonal subspaces. Thus, for every formula_76 and formula_77 in formula_38, formula_79. Equivalently:
Hilbert transform
The Hilbert transform is an anti-self adjoint operator relative to the duality pairing between "L"(R) and the dual space "L"(R), where "p" and "q" are Hölder conjugates and 1 Banach space of "real"-valued functions in "L"(R), the Hilbert transform defines a linear complex structure on this Banach space. In particular, when "p"=2, the Hilbert transform gives the Hilbert space of real-valued functions in "L"(R) the structure of a "complex" Hilbert space.
Hilbert symbol
The Hilbert symbol can also be used to denote the central simple algebra over "K" with basis 1,"i","j","k" and multiplication rules formula_4, formula_5, formula_6. In this case the algebra represents an element of order 2 in the Brauer group of "K", which is identified with -1 if it is a division algebra and +1 if it is isomorphic to the algebra of 2 by 2 matrices.
Hilbert space
Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term "Hilbert space" for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions.
Projective Hilbert space
In mathematics and the foundations of quantum mechanics, the projective Hilbert space formula_1 of a complex Hilbert space formula_2 is the set of equivalence classes of vectors formula_3 in formula_2, with formula_5, for the relation formula_6 given by
Dirichlet space
In mathematics, the Dirichlet space on the domain formula_1 (named after Peter Gustav Lejeune Dirichlet), is the reproducing kernel Hilbert space of holomorphic functions, contained within the Hardy space formula_2, for which the "Dirichlet integral", defined by
Hilbert C*-module
A complex Hilbert space "H" is a Hilbert C-module under its inner product, the complex numbers being a C*-algebra with an involution given by complex conjugation.
Hilbert's Nullstellensatz
Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem"—see "Satz") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry, a branch of mathematics. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. This relationship was discovered by David Hilbert who proved the Nullstellensatz and several other important related theorems named after him (like Hilbert's basis theorem).
Beckman–Quarles theorem
Beckman and Quarles also provide a counterexample for Hilbert space, the space of square-summable sequences of real numbers. This example involves the composition of two discontinuous functions: one that maps every point of the Hilbert space onto a nearby point in a countable dense subspace, and a second that maps this dense set into a countable unit simplex (an infinite set of points all at unit distance from each other). These two transformations map any two points at unit distance from each other to two different points in the dense subspace, and from there map them to two different points of the simplex, which are necessarily at unit distance apart. Therefore, their composition preserves unit distances. However, it is not an isometry, because it maps every pair of points, no matter their original distance, either to the same point or to a unit distance.
Hilbert space
of complex numbers such that
David Hilbert
Additionally, Hilbert's work anticipated and assisted several advances in the mathematical formulation of quantum mechanics. His work was a key aspect of Hermann Weyl and John von Neumann's work on the mathematical equivalence of Werner Heisenberg's matrix mechanics and Erwin Schrödinger's wave equation and his namesake Hilbert space plays an important part in quantum theory. In 1926 von Neumann showed that if atomic states were understood as vectors in Hilbert space, then they would correspond with both Schrödinger's wave function theory and Heisenberg's matrices.
Triangular matrix
The fact that commuting matrices have a common eigenvector can be interpreted as a result of Hilbert's Nullstellensatz: commuting matrices form a commutative algebra formula_21 over formula_22 which can be interpreted as a variety in "k"-dimensional affine space, and the existence of a (common) eigenvalue (and hence a common eigenvector) corresponds to this variety having a point (being non-empty), which is the content of the (weak) Nullstellensatz. In algebraic terms, these operators correspond to an algebra representation of the polynomial algebra in "k" variables.
Frame (linear algebra)
In signal processing, it is common to assume the vector space is a Hilbert space.
Liouville space
In mathematics, a Liouville space, also known as a line space or an extended Hilbert space is Cartesian product of two Hilbert spaces.
Hilbert space
The adjoint of a densely defined unbounded operator is defined in essentially the same manner as for bounded operators. Self-adjoint unbounded operators play the role of the "observables" in the mathematical formulation of quantum mechanics. Examples of self-adjoint unbounded operators on the Hilbert space are:These correspond to the momentum and position observables, respectively. Note that neither nor is defined on all of , since in the case of the derivative need not exist, and in the case of the product function need not be square integrable. In both cases, the set of possible arguments form dense subspaces of .
Hilbert manifold
In mathematics, a Hilbert manifold is a manifold modeled on Hilbert spaces. Thus it is a separable Hausdorff space in which each point has a neighbourhood homeomorphic to an infinite dimensional Hilbert space. The concept of a Hilbert manifold provides a possibility of extending the theory of manifolds to infinite-dimensional setting. Analogously to the finite-dimensional situation, one can define a "differentiable" Hilbert manifold by considering a maximal atlas in which the transition maps are differentiable.
Hypersurface
One of the main such properties is Hilbert's Nullstellensatz, which asserts that a hypersurface contains an algebraic set if and only if the defining polynomial of the hypersurface has a power that belongs to the ideal generated by the defining polynomials of the algebraic set.
Hilbert's inequality
and
Hilbert space
where the dot indicates the dot product in the Euclidean space of partial derivatives of each order. Sobolev spaces can also be defined when is not an integer.
Weak convergence (Hilbert space)
The Hilbert space formula_14 is the space of the square-integrable functions on the interval formula_15 equipped with the inner product defined by
(see L space). The sequence of functions formula_17 defined by
converges weakly to the zero function in formula_14, as the integral
tends to zero for any square-integrable function formula_21 on formula_15 when formula_23 goes to infinity, which is by Riemann–Lebesgue lemma, i.e.
Although formula_25 has an increasing number of 0's in formula_26 as formula_23 goes to infinity, it is of course not equal to the zero function for any formula_23. Note that formula_25 does not converge to 0 in the formula_30 or formula_31 norms. This dissimilarity is one of the reasons why this type of convergence is considered to be "weak."
Space (mathematics)
The most detailed information was carried by a class of spaces called Banach algebras. These are Banach spaces together with a continuous multiplication operation. An important early example was the Banach algebra of essentially bounded measurable functions on a measure space "X". This set of functions is a Banach space under pointwise addition and scalar multiplication. With the operation of pointwise multiplication, it becomes a special type of Banach space, one now called a commutative von Neumann algebra. Pointwise multiplication determines a representation of this algebra on the Hilbert space of square integrable functions on "X". An early observation of John von Neumann was that this correspondence also worked in reverse: Given some mild technical hypotheses, a commutative von Neumann algebra together with a representation on a Hilbert space determines a measure space, and these two constructions (of a von Neumann algebra plus a representation and of a measure space) are mutually inverse.
Hilbert space
By definition, every Hilbert space is also a Banach space. Furthermore, in every Hilbert space the following parallelogram identity holds:
Conversely, every Banach space in which the parallelogram identity holds is a Hilbert space, and the inner product is uniquely determined by the norm by the polarization identity. For real Hilbert spaces, the polarization identity is
For complex Hilbert spaces, it is
The parallelogram law implies that any Hilbert space is a uniformly convex Banach space.
O*-algebra
In mathematics, an O*-algebra is an algebra of possibly unbounded operators defined on a dense subspace of a Hilbert space. The original examples were described by and , who studied some examples of O*-algebras, called Borchers algebras, arising from the Wightman axioms of quantum field theory. and began the systematic study of algebras of unbounded operators.
Direct sum of modules
This space is complete and we get a Hilbert space.
Strong subadditivity of quantum entropy
Suppose that the Hilbert space of the system is a tensor product of three spaces: formula_26. Physically, these three spaces can
be interpreted as the space of three different systems, or else as three parts or three degrees of freedom
of one physical system.
Hilbert's inequality
where
Hilbert space
(according to the definition of the sum of an arbitrary family of non-negative real numbers).
Hilbert series and Hilbert polynomial
Thus formula_62 is an Artinian ring, which is a -vector space of dimension , and Jordan–Hölder theorem may be used for proving that formula_63 is the degree of the algebraic set . In fact, the multiplicity of a point is the number of occurrences of the corresponding maximal ideal in a composition series.
Representation of a Lie superalgebra
A unitary representation of such a Lie algebra is a Z graded Hilbert space which is a representation of a Lie superalgebra as above together with the requirement that self-adjoint elements of the Lie superalgebra are represented by Hermitian transformations.
Abelian von Neumann algebra
Though there is a theory for von Neumann algebras on non-separable Hilbert spaces (and indeed much of the general theory still holds in that case) the theory is considerably simpler for algebras on separable spaces and most applications to other areas of mathematics or physics only use separable Hilbert spaces. Note that if the measure spaces ("X", μ) is a standard measure space (that is "X" − "N" is a standard Borel space for some null set "N" and μ is a σ-finite measure) then "L"("X", μ) is separable.
Hilbert scheme
Pick a sufficiently large value of . The -dimensional space is a subspace of the -dimensional space , so represents a point of the Grassmannian . This will give an embedding of the piece of the Hilbert scheme corresponding to the Hilbert polynomial into this Grassmannian.
Commutation theorem
A Hilbert algebra is an algebra formula_23 with involution "x"→"x"* and an inner product (,) such that
Let "H" be the Hilbert space completion of formula_23 with respect to the inner product and let "J" denote the extension of the involution to a conjugate-linear involution of "H". Define a representation λ and an anti-representation ρ of 
formula_23 on itself by left and right multiplication:
Hilbert metric
In mathematics, the Hilbert metric, also known as the Hilbert projective metric, is an explicitly defined distance function on a bounded convex subset of the "n"-dimensional Euclidean space R. It was introduced by as a generalization of the Cayley's formula for the distance in the Cayley–Klein model of hyperbolic geometry, where the convex set is the "n"-dimensional open unit ball. Hilbert's metric has been applied to Perron–Frobenius theory and to constructing Gromov hyperbolic spaces.
Segal–Bargmann space
where here "dz" denotes the 2"n"-dimensional Lebesgue measure on "C". It is a Hilbert space with respect to the associated inner product:
C*-algebra
C-algebras (pronounced "C-star") are subjects of research in functional analysis, a branch of mathematics. A C*-algebra is a complex algebra "A" of continuous linear operators on a complex Hilbert space with two additional properties:C*-algebras were first considered primarily for their use in quantum mechanics to model algebras of physical observables. This line of research began with Werner Heisenberg's matrix mechanics and in a more mathematically developed form with Pascual Jordan around 1933. Subsequently, John von Neumann attempted to establish a general framework for these algebras which culminated in a series of papers on rings of operators. These papers considered a special class of C*-algebras which are now known as von Neumann algebras.
Hilbert's fourteenth problem
gave the following counterexample to Hilbert's problem. The field "k" is a field containing 48 elements "a", ...,"a", for "i"=1, 2, 3 that are algebraically independent over the prime field. The ring "R" is the polynomial ring "k"["x"...,"x", "t"...,"t"] in 32 variables. The vector space "V" is a 13-dimensional vector space over "k" consisting of all vectors ("b"...,"b") in "k" orthogonal to each of the three vectors ("a", ...,"a") for "i"=1, 2, 3. The vector space "V" is a 13-dimensional commutative unipotent algebraic group under addition, and its elements act on "R" by fixing all elements "t" and taking "x" to "x" + "b""t". Then the ring of elements of "R" invariant under the action of the group "V" is not a finitely generated "k"-algebra.
Hilbert space
The set of all bounded linear operators on (operators ), together with the addition and composition operations, the norm and the adjoint operation, is a C*-algebra, which is a type of operator algebra.
Hilbert space
This subsection employs the Hilbert projection theorem. If is a non-empty closed convex subset of a Hilbert space and a point in , there exists a unique point that minimizes the distance between and points in ,
Hilbert space
Prior to the development of Hilbert spaces, other generalizations of Euclidean spaces were known to mathematicians and physicists. In particular, the idea of an abstract linear space had gained some traction towards the end of the 19th century: this is a space whose elements can be added together and multiplied by scalars (such as real or complex numbers) without necessarily identifying these elements with "geometric" vectors, such as position and momentum vectors in physical systems. Other objects studied by mathematicians at the turn of the 20th century, in particular spaces of sequences (including series) and spaces of functions, can naturally be thought of as linear spaces. Functions, for instance, can be added together or multiplied by constant scalars, and these operations obey the algebraic laws satisfied by addition and scalar multiplication of spatial vectors.
Vector space
Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces, whose vectors are functions. These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are more commonly used, as having a notion of distance between two vectors. This is particularly the case of Banach spaces and Hilbert spaces, which are fundamental in mathematical analysis.
Rigged Hilbert space
is an eigenfunction of the differential operator
Abelian von Neumann algebra
In functional analysis, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute.
Hilbert space
The spectral theorem for compact self-adjoint operators on a Hilbert space states that splits into an orthogonal direct sum of the eigenspaces of an operator, and also gives an explicit decomposition of the operator as a sum of projections onto the eigenspaces. The direct sum of Hilbert spaces also appears in quantum mechanics as the Fock space of a system containing a variable number of particles, where each Hilbert space in the direct sum corresponds to an additional degree of freedom for the quantum mechanical system. In representation theory, the Peter–Weyl theorem guarantees that any unitary representation of a compact group on a Hilbert space splits as the direct sum of finite-dimensional representations.
Geometric quantization
The next step in the process of geometric quantization is the choice of a polarization. A polarization is a choice at each point in formula_6 a Lagrangian subspace of the complexified tangent space of formula_6. The subspaces should form an integrable distribution, meaning that the commutator of two vector fields lying in the subspace at each point should also lie in the vector field at each point. The "quantum" (as opposed to prequantum) Hilbert space is the space of sections of formula_14 that are covariantly constant in the direction of the polarization. The idea is that in the quantum Hilbert space, the sections should be functions of only formula_27 variables on the formula_28-dimensional classical phase space.
Hilbert cube
Every G-subset of the Hilbert cube is a Polish space, a topological space homeomorphic to a separable and complete metric space. Conversely, every Polish space is homeomorphic to a G-subset of the Hilbert cube.
Von Neumann algebra
The Hilbert space tensor product of two Hilbert spaces is the completion of their algebraic tensor product. One can define a tensor product of von Neumann algebras (a completion of the algebraic tensor product of the algebras considered as rings), which is again a von Neumann algebra, and act on the tensor product of the corresponding Hilbert spaces. The tensor product of two finite algebras is finite, and the tensor product of an infinite algebra and a non-zero algebra is infinite. The type of the tensor product of two von Neumann algebras (I, II, or III) is the maximum of their types. The commutation theorem for tensor products states that
Hilbert space
As a complete normed space, Hilbert spaces are by definition also Banach spaces. As such they are topological vector spaces, in which topological notions like the openness and closedness of subsets are well defined. Of special importance is the notion of a closed linear subspace of a Hilbert space that, with the inner product induced by restriction, is also complete (being a closed set in a complete metric space) and therefore a Hilbert space in its own right.
Hilbert transform
Thus
Hilbert matrix
where
Uniformly hyperfinite algebra
One example of a UHF C*-algebra is the CAR algebra. It is defined as follows: let "H" be a separable complex Hilbert space "H" with orthonormal basis "f" and "L"("H") the bounded operators on "H", consider a linear map
Hilbert space
The problem can also be studied from the abstract point of view: every Hilbert space has an orthonormal basis, and every element of the Hilbert space can be written in a unique way as a sum of multiples of these basis elements. The coefficients appearing on these basis elements are sometimes known abstractly as the Fourier coefficients of the element of the space. The abstraction is especially useful when it is more natural to use different basis functions for a space such as . In many circumstances, it is desirable not to decompose a function into trigonometric functions, but rather into orthogonal polynomials or wavelets for instance, and in higher dimensions into spherical harmonics.
Hilbert space
Sobolev spaces, denoted by or , are Hilbert spaces. These are a special kind of function space in which differentiation may be performed, but that (unlike other Banach spaces such as the Hölder spaces) support the structure of an inner product. Because differentiation is permitted, Sobolev spaces are a convenient setting for the theory of partial differential equations. They also form the basis of the theory of direct methods in the calculus of variations.
Semi-Hilbert space
In mathematics, a semi-Hilbert space is a generalization of a Hilbert space in functional analysis, in which, roughly speaking, the inner product is required only to be positive semi-definite rather than positive definite, so that it gives rise to a seminorm rather than a vector space norm.
Hilbert space
Further basic results were proved in the early 20th century. For example, the Riesz representation theorem was independently established by Maurice Fréchet and Frigyes Riesz in 1907. John von Neumann coined the term "abstract Hilbert space" in his work on unbounded Hermitian operators. Although other mathematicians such as Hermann Weyl and Norbert Wiener had already studied particular Hilbert spaces in great detail, often from a physically motivated point of view, von Neumann gave the first complete and axiomatic treatment of them. Von Neumann later used them in his seminal work on the foundations of quantum mechanics, and in his continued work with Eugene Wigner. The name "Hilbert space" was soon adopted by others, for example by Hermann Weyl in his book on quantum mechanics and the theory of groups.
Crossed product
Suppose that "A" is a von Neumann algebra of operators acting on a Hilbert space "H" and "G" is a discrete group acting on "A". We let "K" be the Hilbert space of all square summable "H"-valued functions on "G". There is an action of "A" on "K"
given by 
for "k" in "K", "g", "h" in "G", and "a" in "A",
and there is an action of "G" on "K" given by
The crossed product formula_5 is the von Neumann algebra acting on "K" generated by the actions of "A" and "G" on "K". It does not depend (up to isomorphism) on the choice of the Hilbert space "H".
Hilbert space
The space of square-summable sequences of complex numbers is the set of infinite sequences
Hilbert series and Hilbert polynomial
The Hilbert series of an algebra or a module is a special case of the Hilbert–Poincaré series of a graded vector space.
Hilbert space
Hilbert spaces are often taken over the complex numbers. The complex plane denoted by is equipped with a notion of magnitude, the complex modulus which is defined as the square root of the product of with its complex conjugate:
If is a decomposition of into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length:
The inner product of a pair of complex numbers and is the product of with the complex conjugate of :
This is complex-valued. The real part of gives the usual two-dimensional Euclidean dot product.
David Hilbert
Around 1909, Hilbert dedicated himself to the study of differential and integral equations; his work had direct consequences for important parts of modern functional analysis. In order to carry out these studies, Hilbert introduced the concept of an infinite dimensional Euclidean space, later called Hilbert space. His work in this part of analysis provided the basis for important contributions to the mathematics of physics in the next two decades, though from an unanticipated direction.
Later on, Stefan Banach amplified the concept, defining Banach spaces. Hilbert spaces are an important class of objects in the area of functional analysis, particularly of the spectral theory of self-adjoint linear operators, that grew up around it during the 20th century.
Sobolev space
Sobolev spaces with are especially important because of their connection with Fourier series and because they form a Hilbert space. A special notation has arisen to cover this case, since the space is a Hilbert space:
Singular trace
In mathematics, a singular trace is a trace on a space of linear operators of a separable Hilbert space that vanishes
on operators of finite rank. Singular traces are a feature of infinite-dimensional Hilbert spaces such as the space of square-summable sequences and spaces of square-integrable functions. Linear operators on a finite-dimensional Hilbert space have only the zero functional as a singular trace since all operators have finite rank. For example, matrix algebras have no non-trivial singular traces and the matrix trace is the unique trace up to scaling.
Hilbert space
A Bergman space is an example of a reproducing kernel Hilbert space, which is a Hilbert space of functions along with a kernel that verifies a reproducing property analogous to this one. The Hardy space also admits a reproducing kernel, known as the Szegő kernel. Reproducing kernels are common in other areas of mathematics as well. For instance, in harmonic analysis the Poisson kernel is a reproducing kernel for the Hilbert space of square-integrable harmonic functions in the unit ball. That the latter is a Hilbert space at all is a consequence of the mean value theorem for harmonic functions.
Hilbert's inequality
and
Rigged Hilbert space
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 bring together the 'bound state' (eigenvector) and 'continuous spectrum', in one place.
C*-algebra
Around 1943, the work of Israel Gelfand and Mark Naimark yielded an abstract characterisation of C*-algebras making no reference to operators on a Hilbert space.
Constraint algebra
In theoretical physics, a constraint algebra is a linear space of all constraints and all of their polynomial functions or functionals whose action on the physical vectors of the Hilbert space should be equal to zero.
Direct integral
where H is the canonical "n"-dimensional Hilbert space, that is
Weak convergence (Hilbert space)
i.e.
Observable
If the Hilbert space is finite-dimension, it can be represented by a Hermitian matrix if the space is finite-dimensional. In an infinite-dimensional Hilbert space, the observable is represented by a symmetric operator, which may not be defined everywhere. The reason for such a change is that in an infinite-dimensional Hilbert space, the observable operator can become unbounded, which means that it no longer has a largest eigenvalue. This is not the case in a finite-dimensional Hilbert space: an operator can have no more eigenvalues than the dimension of the state it acts upon, and by the well-ordering property, any finite set of real numbers has a largest element. For example, the position of a point particle moving along a line can take any real number as its value, and the set of real numbers is uncountably infinite. Since the eigenvalue of an observable represents a possible physical quantity that its corresponding dynamical variable can take, we must conclude that there is no largest eigenvalue for the position observable in this uncountably infinite-dimensional Hilbert space. 
A crucial difference between classical quantities and quantum mechanical observables is that the latter may not be simultaneously measurable, a property referred to as complementarity. This is mathematically expressed by non-commutativity of the corresponding operators, to the effect that the commutator
Hilbert metric
Let Ω be a convex open domain in a Euclidean space that does not contain a line. Given two distinct points "A" and "B" of Ω, let "X" and "Y" be the points at which the straight line "AB" intersects the boundary of Ω, where the order of the points is "X", "A", "B", "Y". Then the Hilbert distance "d"("A", "B") is the logarithm of the cross-ratio of this quadruple of points:
Constraint algebra
For example, in electromagnetism, the equation for the Gauss' law
is an equation of motion that does not include any time derivatives. This is why it is counted as a constraint, not a dynamical equation of motion. In quantum electrodynamics, one first constructs a Hilbert space in which Gauss' law does not hold automatically. The true Hilbert space of physical states is constructed as a subspace of the original Hilbert space of vectors that satisfy
In more general theories, the constraint algebra may be a noncommutative algebra.
Hilbert space
Lebesgue spaces are function spaces associated to measure spaces , where is a set, is a σ-algebra of subsets of , and is a countably additive measure on . Let be the space of those complex-valued measurable functions on for which the Lebesgue integral of the square of the absolute value of the function is finite, i.e., for a function in ,
Hilbert space
for all and .
Hilbert's arithmetic of ends
In mathematics, specifically in the area of hyperbolic geometry, Hilbert's arithmetic of ends is a method for endowing a geometric set, the set of ideal points or "ends" of a hyperbolic plane, with an algebraic structure as a field.
It was introduced by German mathematician David Hilbert.
Multiplier algebra
For example, if "A" is the C*-algebra of compact operators on a separable Hilbert space, "M"("A") is "B"("H"), the C*-algebra of all bounded operators on "H".
Hilbert space
Unbounded operators are also tractable in Hilbert spaces, and have important applications to quantum mechanics. An unbounded operator on a Hilbert space is defined as a linear operator whose domain is a linear subspace of . Often the domain is a dense subspace of , in which case is known as a densely defined operator.
Hilbert space
As a consequence of Parseval's identity, if is an orthonormal basis of , then the map defined by is an isometric isomorphism of Hilbert spaces: it is a bijective linear mapping such that
for all . The cardinal number of is the Hilbert dimension of . Thus every Hilbert space is isometrically isomorphic to a sequence space for some set .
Hilbert space
the supremum being taken over all finite subsets of . It follows that, for this sum to be finite, every element of has only countably many nonzero terms. This space becomes a Hilbert space with the inner product
for all . Here the sum also has only countably many nonzero terms, and is unconditionally convergent by the Cauchy–Schwarz inequality.
Hilbert series and Hilbert polynomial
The ring formula_51 has Krull dimension one, and is the ring of regular functions of a projective algebraic set formula_52 of dimension 0 consisting of a finite number of points, which may be multiple points. As formula_53 belongs to a regular sequence, none of these points belong to the hyperplane of equation formula_54 The complement of this hyperplane is an affine space that contains formula_55 This makes formula_52 an affine algebraic set, which has formula_57 as its ring of regular functions. The linear polynomial formula_58 is not a zero divisor in formula_59 and one has thus an exact sequence
which implies that 
Here we are using Hilbert series of filtered algebras, and the fact that the Hilbert series of a graded algebra is also its Hilbert series as filtered algebra.
Hilbert algebra
In mathematics, Hilbert algebras occur in the theory of von Neumann algebras in:
Commutator subspace
Commutators of linear operators on Hilbert spaces came to prominence in the 1930s as they featured in the matrix mechanics, or Heisenberg, formulation of quantum mechanics. Commutator subspaces, though, received sparse attention until the 1970s. American mathematician Paul Halmos in 1954 showed that every bounded operator on a separable infinite dimensional Hilbert space is the sum of two commutators of bounded operators.
In 1971 Carl Pearcy and David Topping revisited the topic and studied commutator subspaces for Schatten ideals. As a student American mathematician Gary Weiss began to investigate spectral conditions for commutators of Hilbert–Schmidt operators.
British mathematician Nigel Kalton, noticing the spectral condition of Weiss, characterised all trace class commutators.
Kalton's result forms the basis for the modern characterisation of the commutator subspace.
In 2004 Ken Dykema, Tadeusz Figiel, Gary Weiss and Mariusz Wodzicki published the spectral characterisation of normal operators in the commutator subspace for every two-sided ideal of compact operators.
Hilbert–Schmidt operator
The Hilbert–Schmidt operators form a two-sided *-ideal in the Banach algebra of bounded operators on "H". They also form a Hilbert space, which can be shown to be naturally isometrically isomorphic to the tensor product of Hilbert spaces
Rigged Hilbert space
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
Hilbert–Schmidt operator
In mathematics, a Hilbert–Schmidt operator, named for David Hilbert and Erhard Schmidt, is a bounded operator "A" on a Hilbert space "H" with finite Hilbert–Schmidt norm
Hilbert series and Hilbert polynomial
The Hilbert function
maps the integer onto the dimension of the -vector space . The Hilbert series, which is called Hilbert–Poincaré series in the more general setting of graded vector spaces, is the formal series