The state of avibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinctovertones is given by the projection of the point onto thecoordinate axes in the space.
Inmathematics, aHilbert space (named forDavid Hilbert) generalizes the notion ofEuclidean space. It extends the methods oflinear algebra andcalculus from the two-dimensionalEuclidean plane and three-dimensional space to spaces with any finite or infinite number ofdimensions. A Hilbert space is avector space equipped with aninner product operation, which allows lengths and angles to be defined. Furthermore, Hilbert spaces arecomplete, which means that there are enoughlimits in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of aBanach space.
One of the most familiar examples of a Hilbert space is theEuclidean vector space consisting of three-dimensionalvectors, denoted byR3, and equipped with thedot product. The dot product takes two vectorsx andy, and produces a real numberx ⋅y. Ifx andy are represented inCartesian coordinates, then the dot product is defined by
An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real)inner product. Avector space equipped with such an inner product is known as a (real)inner product space. Every finite-dimensional inner product space is also a Hilbert space.[2] The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length (ornorm) of a vector, denoted‖x‖, and to the angleθ between two vectorsx andy by means of the formula
Completeness means that a series of vectors (in blue) results in awell-defined net displacement vector (in orange).
Multivariable calculus in Euclidean space relies on the ability to computelimits, and to have useful criteria for concluding that limits exist. Amathematical seriesconsisting of vectors inR3 isabsolutely convergent provided that the sum of the lengths converges as an ordinary series of real numbers:[3]
Just as with a series of scalars, a series of vectors that converges absolutely also converges to some limit vectorL in the Euclidean space, in the sense that
This property expresses thecompleteness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense.
Ifz =x +iy is a decomposition ofz into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length:
The inner product of a pair of complex numbersz andw is the product ofz with the complex conjugate ofw:
This is complex-valued. The real part of⟨z,w⟩ gives the usual two-dimensional Euclideandot product.
A second example is the spaceC2 whose elements are pairs of complex numbersz = (z1,z2). Then an inner product ofz with another such vectorw = (w1,w2) is given by
The real part of⟨z,w⟩ is then the four-dimensional Euclidean dot product. This inner product isHermitian symmetric, which means that the result of interchangingz andw is the complex conjugate:
To say that a complex vector spaceH is acomplex inner product space means that there is an inner product associating a complex number to each pair of elements ofH that satisfies the following properties:
The inner product is conjugate symmetric; that is, the inner product of a pair of elements is equal to thecomplex conjugate of the inner product of the swapped elements: Importantly, this implies that is a real number.
The inner product islinear in its first[nb 1] argument. For all complex numbers and
It follows from properties 1 and 2 that a complex inner product isantilinear, also calledconjugate linear, in its second argument, meaning that
Areal inner product space is defined in the same way, except thatH is a real vector space and the inner product takes real values. Such an inner product will be abilinear map and will form adual system.[5]
Thenorm is the real-valued functionand the distance between two points inH is defined in terms of the norm by
That this function is a distance function means firstly that it is symmetric in and secondly that the distance between and itself is zero, and otherwise the distance between and must be positive, and lastly that thetriangle inequality holds, meaning that the length of one leg of a trianglexyz cannot exceed the sum of the lengths of the other two legs:
With a distance function defined in this way, any inner product space is ametric space, and sometimes is known as apre-Hilbert space.[6] Any pre-Hilbert space that is additionally also acomplete space is a Hilbert space.[7]
As a complete normed space, Hilbert spaces are by definition alsoBanach spaces. As such they aretopological vector spaces, in whichtopological notions like theopenness andclosedness of subsets arewell defined. Of special importance is the notion of a closedlinear subspace of a Hilbert space that, with the inner product induced byrestriction, is also complete (being a closed set in a complete metric space) and therefore a Hilbert space in its own right.
This second series converges as a consequence of theCauchy–Schwarz inequality and the convergence of the previous series.
Completeness of the space holds provided that whenever a series of elements froml2 converges absolutely (in norm), then it converges to an element ofl2. The proof is basic inmathematical analysis, and permits mathematical series of elements of the space to be manipulated with the same ease as series of complex numbers (or vectors in a finite-dimensional Euclidean space).[10]
Prior to the development of Hilbert spaces, other generalizations of Euclidean spaces were known tomathematicians andphysicists. In particular, the idea of anabstract linear space (vector space) had gained some traction towards the end of the 19th century:[11] this is a space whose elements can be added together and multiplied by scalars (such asreal orcomplex 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 ofsequences (includingseries) and spaces of functions,[12] 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.
In the first decade of the 20th century, parallel developments led to the introduction of Hilbert spaces. The first of these was the observation, which arose duringDavid Hilbert andErhard Schmidt's study ofintegral equations,[13] that twosquare-integrable real-valued functionsf andg on an interval[a,b] have aninner product
that has many of the familiar properties of the Euclidean dot product. In particular, the idea of anorthogonal family of functions has meaning. Schmidt exploited the similarity of this inner product with the usual dot product to prove an analog of thespectral decomposition for an operator of the form
whereK is a continuous function symmetric inx andy. The resultingeigenfunction expansion expresses the functionK as a series of the form
where the functionsφn are orthogonal in the sense that⟨φn,φm⟩ = 0 for alln ≠m. The individual terms in this series are sometimes referred to as elementary product solutions. However, there are eigenfunction expansions that fail to converge in a suitable sense to a square-integrable function: the missing ingredient, which ensures convergence, is completeness.[14]
Further basic results were proved in the early 20th century. For example, theRiesz representation theorem was independently established byMaurice Fréchet andFrigyes Riesz in 1907.[18]John von Neumann coined the termabstract Hilbert space in his work on unboundedHermitian operators.[19] Although other mathematicians such asHermann Weyl andNorbert 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.[20] Von Neumann later used them in his seminal work on the foundations of quantum mechanics,[21] and in his continued work withEugene 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.[22]
The algebra ofobservables in quantum mechanics is naturally an algebra of operators defined on a Hilbert space, according toWerner Heisenberg'smatrix mechanics formulation of quantum theory.[25] Von Neumann began investigatingoperator algebras in the 1930s, asrings of operators on a Hilbert space. The kind of algebras studied by von Neumann and his contemporaries are now known asvon Neumann algebras.[26] In the 1940s,Israel Gelfand,Mark Naimark andIrving Segal gave a definition of a kind of operator algebras calledC*-algebras that on the one hand made no reference to an underlying Hilbert space, and on the other extrapolated many of the useful features of the operator algebras that had previously been studied. The spectral theorem for self-adjoint operators in particular that underlies much of the existing Hilbert space theory was generalized to C*-algebras.[27] These techniques are now basic in abstract harmonic analysis and representation theory.
The inner product of functionsf andg inL2(X,μ) is then defined as or
where the second form (conjugation of the first element) is commonly found in thetheoretical physics literature. Forf andg inL2, the integral exists because of the Cauchy–Schwarz inequality, and defines an inner product on the space. Equipped with this inner product,L2 is in fact complete.[28] The Lebesgue integral is essential to ensure completeness: on domains of real numbers, for instance, not enough functions areRiemann integrable.[29]
The Lebesgue spaces appear in many natural settings. The spacesL2(R) andL2([0,1]) of square-integrable functions with respect to theLebesgue measure on the real line and unit interval, respectively, are natural domains on which to define the Fourier transform and Fourier series. In other situations, the measure may be something other than the ordinary Lebesgue measure on the real line. For instance, ifw is any positive measurable function, the space of all measurable functionsf on the interval[0, 1] satisfyingis called theweightedL2 spaceL2 w([0, 1]), andw is called the weight function. The inner product is defined by
The weighted spaceL2 w([0, 1]) is identical with the Hilbert spaceL2([0, 1],μ) where the measureμ of a Lebesgue-measurable setA is defined by
WeightedL2 spaces like this are frequently used to studyorthogonal polynomials, because different families of orthogonal polynomials are orthogonal with respect to different weighting functions.[30]
Fors a non-negativeinteger andΩ ⊂Rn, the Sobolev spaceHs(Ω) containsL2 functions whoseweak derivatives of order up tos are alsoL2. The inner product inHs(Ω) iswhere the dot indicates the dot product in the Euclidean space of partial derivatives of each order. Sobolev spaces can also be defined whens is not an integer.
Sobolev spaces are also studied from the point of view of spectral theory, relying more specifically on the Hilbert space structure. IfΩ is a suitable domain, then one can define the Sobolev spaceHs(Ω) as the space ofBessel potentials;[33] roughly,
TheHardy spaces are function spaces, arising incomplex analysis andharmonic analysis, whose elements are certainholomorphic functions in a complex domain.[35] LetU denote theunit disc in the complex plane. Then the Hardy spaceH2(U) is defined as the space of holomorphic functionsf onU such that the meansremain bounded forr < 1. The norm on this Hardy space is defined by
Hardy spaces in the disc are related to Fourier series. A functionf is inH2(U) if and only ifwhere
ThusH2(U) consists of those functions that areL2 on the circle, and whose negative frequency Fourier coefficients vanish.
TheBergman spaces are another family of Hilbert spaces of holomorphic functions.[36] LetD be a bounded open set in thecomplex plane (or a higher-dimensional complex space) and letL2,h(D) be the space of holomorphic functionsf inD that are also inL2(D) in the sense thatwhere the integral is taken with respect to the Lebesgue measure inD. ClearlyL2,h(D) is a subspace ofL2(D); in fact, it is aclosed subspace, and so a Hilbert space in its own right. This is a consequence of the estimate, valid oncompact subsetsK ofD, thatwhich in turn follows fromCauchy's integral formula. Thus convergence of a sequence of holomorphic functions inL2(D) implies alsocompact convergence, and so the limit function is also holomorphic. Another consequence of this inequality is that the linear functional that evaluates a functionf at a point ofD is actually continuous onL2,h(D). The Riesz representation theorem implies that the evaluation functional can be represented as an element ofL2,h(D). Thus, for everyz ∈D, there is a functionηz ∈L2,h(D) such thatfor allf ∈L2,h(D). The integrandis known as theBergman kernel ofD. Thisintegral kernel satisfies a reproducing property
A Bergman space is an example of areproducing kernel Hilbert space, which is a Hilbert space of functions along with a kernelK(ζ,z) that verifies a reproducing property analogous to this one. The Hardy spaceH2(D) also admits a reproducing kernel, known as theSzegő kernel.[37] Reproducing kernels are common in other areas of mathematics as well. For instance, inharmonic analysis thePoisson kernel is a reproducing kernel for the Hilbert space of square-integrableharmonic functions in theunit ball. That the latter is a Hilbert space at all is a consequence of the mean value theorem for harmonic functions.
Many of the applications of Hilbert spaces exploit the fact that Hilbert spaces support generalizations of simple geometric concepts likeprojection andchange of basis from their usual finite dimensional setting. In particular, thespectral theory ofcontinuousself-adjointlinear operators on a Hilbert space generalizes the usualspectral decomposition of amatrix, and this often plays a major role in applications of the theory to other areas of mathematics and physics.
Theovertones of a vibrating string. These areeigenfunctions of an associated Sturm–Liouville problem. The eigenvalues 1,1/2,1/3, ... form the (musical)harmonic series.
In the theory ofordinary differential equations, spectral methods on a suitable Hilbert space are used to study the behavior of eigenvalues and eigenfunctions of differential equations. For example, theSturm–Liouville problem arises in the study of the harmonics of waves in a violin string or a drum, and is a central problem inordinary differential equations.[38] The problem is a differential equation of the formfor an unknown functiony on an interval[a,b], satisfying general homogeneousRobin boundary conditionsThe functionsp,q, andw are given in advance, and the problem is to find the functiony and constantsλ for which the equation has a solution. The problem only has solutions for certain values ofλ, called eigenvalues of the system, and this is a consequence of the spectral theorem forcompact operators applied to theintegral operator defined by theGreen's function for the system. Furthermore, another consequence of this general result is that the eigenvaluesλ of the system can be arranged in an increasing sequence tending to infinity.[39][nb 2]
Hilbert spaces form a basic tool in the study ofpartial differential equations.[31] For many classes of partial differential equations, such as linearelliptic equations, it is possible to consider a generalized solution (known as aweak solution) by enlarging the class of functions. Many weak formulations involve the class ofSobolev functions, which is a Hilbert space. A suitable weak formulation reduces to a geometrical problem, the analytic problem of finding a solution or, often what is more important, showing that a solution exists and is unique for given boundary data. For linear elliptic equations, one geometrical result that ensures unique solvability for a large class of problems is theLax–Milgram theorem. This strategy forms the rudiment of theGalerkin method (afinite element method) for numerical solution of partial differential equations.[40]
A typical example is thePoisson equation−Δu =g withDirichlet boundary conditions in a bounded domainΩ inR2. The weak formulation consists of finding a functionu such that, for all continuously differentiable functionsv inΩ vanishing on the boundary:
This can be recast in terms of the Hilbert spaceH1 0(Ω) consisting of functionsu such thatu, along with its weak partial derivatives, are square integrable onΩ, and vanish on the boundary. The question then reduces to findingu in this space such that for allv in this space
Since the Poisson equation iselliptic, it follows from Poincaré's inequality that the bilinear forma iscoercive. The Lax–Milgram theorem then ensures the existence and uniqueness of solutions of this equation.[41]
The field ofergodic theory is the study of the long-term behavior ofchaoticdynamical systems. The protypical case of a field that ergodic theory applies to isthermodynamics, in which—though the microscopic state of a system is extremely complicated (it is impossible to understand the ensemble of individual collisions between particles of matter)—the average behavior over sufficiently long time intervals is tractable. Thelaws of thermodynamics are assertions about such average behavior. In particular, one formulation of thezeroth law of thermodynamics asserts that over sufficiently long timescales, the only functionally independent measurement that one can make of a thermodynamic system in equilibrium is its total energy, in the form oftemperature.[43]
An ergodic dynamical system is one for which, apart from the energy—measured by theHamiltonian—there are no other functionally independentconserved quantities on thephase space. More explicitly, suppose that the energyE is fixed, and letΩE be the subset of the phase space consisting of all states of energyE (an energy surface), and letTt denote the evolution operator on the phase space. The dynamical system is ergodic if every invariant measurable functions onΩE is constantalmost everywhere.[44] An invariant functionf is one for whichfor allw onΩE and all timet.Liouville's theorem implies that there exists ameasureμ on the energy surface that is invariant under thetime translation. As a result, time translation is aunitary transformation of the Hilbert spaceL2(ΩE,μ) consisting of square-integrable functions on the energy surfaceΩE with respect to the inner product
The von Neumann mean ergodic theorem[24] states the following:
IfUt is a (strongly continuous) one-parametersemigroup of unitary operators on a Hilbert spaceH, andP is the orthogonal projection onto the space of common fixed points ofUt,{x ∈H |Utx =x, ∀t > 0}, then
For an ergodic system, the fixed set of the time evolution consists only of the constant functions, so the ergodic theorem implies the following:[45] for any functionf ∈L2(ΩE,μ),
That is, the long time average of an observablef is equal to its expectation value over an energy surface.
Superposition of sinusoidal wave basis functions (bottom) to form a sawtooth wave (top)Spherical harmonics, an orthonormal basis for the Hilbert space of square-integrable functions on the sphere, shown graphed along the radial direction
One of the basic goals ofFourier analysis is to decompose a function into a (possibly infinite)linear combination of given basis functions: the associatedFourier series. The classical Fourier series associated to a functionf defined on the interval[0, 1] is a series of the formwhere
The example of adding up the first few terms in a Fourier series for a sawtooth function is shown in the figure. The basis functions are sine waves with wavelengthsλ/n (for integern) shorter than the wavelengthλ of the sawtooth itself (except forn = 1, thefundamental wave).
A significant problem in classical Fourier series asks in what sense the Fourier series converges, if at all, to the functionf. Hilbert space methods provide one possible answer to this question.[46] The functionsen(θ) =e2πinθ form an orthogonal basis of the Hilbert spaceL2([0, 1]). Consequently, any square-integrable function can be expressed as a seriesand, moreover, this series converges in the Hilbert space sense (that is, in theL2 mean).
The problem can also be studied from the abstract point of view: every Hilbert space has anorthonormal 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.[47] The abstraction is especially useful when it is more natural to use different basis functions for a space such asL2([0, 1]). In many circumstances, it is desirable not to decompose a function into trigonometric functions, but rather intoorthogonal polynomials orwavelets for instance,[48] and in higher dimensions intospherical harmonics.[49]
For instance, ifen are any orthonormal basis functions ofL2[0, 1], then a given function inL2[0, 1] can be approximated as a finite linear combination[50]
The coefficients{aj} are selected to make the magnitude of the difference‖f −fn‖2 as small as possible. Geometrically, thebest approximation is theorthogonal projection off onto the subspace consisting of all linear combinations of the{ej}, and can be calculated by[51]
In various applications to physical problems, a function can be decomposed into physically meaningfuleigenfunctions of adifferential operator (typically theLaplace operator): this forms the foundation for the spectral study of functions, in reference to thespectrum of the differential operator.[52] A concrete physical application involves the problem ofhearing the shape of a drum: given the fundamental modes of vibration that a drumhead is capable of producing, can one infer the shape of the drum itself?[53] The mathematical formulation of this question involves theDirichlet eigenvalues of the Laplace equation in the plane, that represent the fundamental modes of vibration in direct analogy with the integers that represent the fundamental modes of vibration of the violin string.
Spectral theory also underlies certain aspects of theFourier transform of a function. Whereas Fourier analysis decomposes a function defined on acompact set into the discrete spectrum of the Laplacian (which corresponds to the vibrations of a violin string or drum), the Fourier transform of a function is the decomposition of a function defined on all of Euclidean space into its components in thecontinuous spectrum of the Laplacian. The Fourier transformation is also geometrical, in a sense made precise by thePlancherel theorem, that asserts that it is anisometry of one Hilbert space (the "time domain") with another (the "frequency domain"). This isometry property of the Fourier transformation is a recurring theme in abstractharmonic analysis (since it reflects the conservation of energy for the continuous Fourier Transform), as evidenced for instance by thePlancherel theorem for spherical functions occurring innoncommutative harmonic analysis.
In the mathematically rigorous formulation ofquantum mechanics, developed byJohn von Neumann,[54] the possible states (more precisely, thepure states) of a quantum mechanical system are represented byunit vectors (calledstate vectors) residing in a complex separable Hilbert space, known as thestate space, well defined up to a complex number of norm 1 (thephase factor). In other words, the possible states are points in theprojectivization of a Hilbert space, usually called thecomplex projective space. The exact nature of this Hilbert space is dependent on the system; for example, the position and momentum states for a single non-relativistic spin zero particle is the space of allsquare-integrable functions, while the states for the spin of a single proton are unit elements of the two-dimensional complex Hilbert space ofspinors. Each observable is represented by aself-adjointlinear operator acting on the state space. Each eigenstate of an observable corresponds to aneigenvector of the operator, and the associatedeigenvalue corresponds to the value of the observable in that eigenstate.[55]
The inner product between two state vectors is a complex number known as aprobability amplitude. During an ideal measurement of a quantum mechanical system, the probability that a system collapses from a given initial state to a particular eigenstate is given by the square of theabsolute value of the probability amplitudes between the initial and final states.[56] The possible results of a measurement are the eigenvalues of the operator—which explains the choice of self-adjoint operators, for all the eigenvalues must be real. The probability distribution of an observable in a given state can be found by computing the spectral decomposition of the corresponding operator.[57]
For a general system, states are typically not pure, but instead are represented as statistical mixtures of pure states, or mixed states, given bydensity matrices: self-adjoint operators oftrace one on a Hilbert space.[58] Moreover, for general quantum mechanical systems, the effects of a single measurement can influence other parts of a system in a manner that is described instead by apositive operator valued measure. Thus the structure both of the states and observables in the general theory is considerably more complicated than the idealization for pure states.[59]
Inprobability theory, Hilbert spaces also have diverse applications. Here a fundamental Hilbert space is the space ofrandom variables on a givenprobability space, having class (finite first and secondmoments). A common operation in statistics is that of centering a random variable by subtracting itsexpectation. Thus if is a random variable, then is its centering. In the Hilbert space view, this is the orthogonal projection of onto thekernel of the expectation operator, which acontinuous linear functional on the Hilbert space (in fact, the inner product with the constant random variable 1), and so this kernel is a closed subspace.
Theconditional expectation has a natural interpretation in the Hilbert space.[60] Suppose that a probability space is given, where is asigma algebra on the set, and is aprobability measure on the measure space. If is a sigma subalgebra of, then the conditional expectation is the orthogonal projection of onto the subspace of consisting of the-measurable functions. If the random variable in is independent of the sigma algebra then conditional expectation, i.e., its projection onto the-measurable functions is constant. Equivalently, the projection of its centering is zero.
In particular, if two random variables and (in) are independent, then the centered random variables and are orthogonal. (This means that the two variables have zerocovariance: they areuncorrelated.) In that case, the Pythagorean theorem in the kernel of the expectation operator implies that thevariances of and satisfy the identity:sometimes called the Pythagorean theorem of statistics, and is of importance inlinear regression.[61] AsStapleton (1995) puts it, "theanalysis of variance may be viewed as the decomposition of the squared length of a vector into the sum of the squared lengths of several vectors, using the Pythagorean Theorem."
The theory ofmartingales can be formulated in Hilbert spaces. A martingale in a Hilbert space is a sequence of elements of a Hilbert space such that, for eachn, is the orthogonal projection of onto the linear hull of.[62] If the are random variables, this reproduces the usual definition of a (discrete) martingale: the expectation of, conditioned on, is equal to.
Hilbert spaces are also used throughout the foundations of theItô calculus.[63] To any square-integrablemartingale, it is possible to associate a Hilbert norm on the space of equivalence classes ofprogressively measurable processes with respect to the martingale (using thequadratic variation of the martingale as the measure). TheItô integral can be constructed by first defining it forsimple processes, and then exploiting their density in the Hilbert space. A noteworthy result is then theItô isometry, which attests that for any martingaleM having quadratic variation measure, and any progressively measurable processH:whenever the expectation on the right-hand side is finite.
A deeper application of Hilbert spaces that is especially important in the theory ofGaussian processes is an attempt, due toLeonard Gross and others, to make sense of certain formal integrals over infinite dimensional spaces like theFeynman path integral fromquantum field theory. The problem with integrals like this is that there is noinfinite dimensional Lebesgue measure. The notion of anabstract Wiener space allows one to construct a measure on a Banach spaceB that contains a Hilbert spaceH, called theCameron–Martin space, as a dense subset, out of a finitely additive cylinder set measure onH. The resulting measure onB is countably additive and invariant under translation by elements ofH, and this provides a mathematically rigorous way of thinking of theWiener measure as a Gaussian measure on the Sobolev space.[64]
Any true physical color can be represented by a combination of purespectral colors. As physical colors can be composed of any number of spectral colors, the space of physical colors may aptly be represented by a Hilbert space over spectral colors. Humans havethree types of cone cells for color perception, so the perceivable colors can be represented by 3-dimensional Euclidean space. The many-to-one linear mapping from the Hilbert space of physical colors to the Euclidean space of human perceivable colors explains why many distinct physical colors may be perceived by humans to be identical (e.g., pure yellow light versus a mix of red and green light, seeMetamerism).[65][66]
Two vectorsu andv in a Hilbert spaceH are orthogonal when⟨u,v⟩ = 0. The notation for this isu ⊥v. More generally, whenS is a subset inH, the notationu ⊥S means thatu is orthogonal to every element fromS.
Whenu andv are orthogonal, one has
By induction onn, this is extended to any familyu1, ...,un ofn orthogonal vectors,
Whereas the Pythagorean identity as stated is valid in any inner product space, completeness is required for the extension of the Pythagorean identity to series.[67] A seriesΣuk oforthogonal vectors converges inH if and only if the series of squares of norms converges, andFurthermore, the sum of a series of orthogonal vectors is independent of the order in which it is taken.
Geometrically, the parallelogram identity asserts thatAC2 + BD2 = 2(AB2 + AD2). In words, the sum of the squares of the diagonals is twice the sum of the squares of any two adjacent sides.
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 thepolarization identity.[69] For real Hilbert spaces, the polarization identity is
This subsection employs theHilbert projection theorem. IfC is a non-empty closed convex subset of a Hilbert spaceH andx a point inH, there exists a unique pointy ∈C that minimizes the distance betweenx and points inC,[71]
This is equivalent to saying that there is a point with minimal norm in the translated convex setD =C −x. The proof consists in showing that every minimizing sequence(dn) ⊂D is Cauchy (using the parallelogram identity) hence converges (using completeness) to a point inD that has minimal norm. More generally, this holds in any uniformly convex Banach space.[72]
When this result is applied to a closed subspaceF ofH, it can be shown that the pointy ∈F closest tox is characterized by[73]
In particular, whenF is not equal toH, one can find a nonzero vectorv orthogonal toF (selectx ∉F andv =x −y). A very useful criterion is obtained by applying this observation to the closed subspaceF generated by a subsetS ofH.
A subsetS ofH spans a dense vector subspace if (and only if) the vector 0 is the sole vectorv ∈H orthogonal toS.
Thedual spaceH* is the space of allcontinuous linear functions from the spaceH into the base field. It carries a natural norm, defined byThis norm satisfies theparallelogram law, and so the dual space is also an inner product space where this inner product can be defined in terms of this dual norm by using thepolarization identity. The dual space is also complete so it is a Hilbert space in its own right. Ife• = (ei)i ∈I is a complete orthonormal basis forH then the inner product on the dual space of any two iswhere all but countably many of the terms in this series are zero.
TheRiesz representation theorem affords a convenient description of the dual space. To every elementu ofH, there is a unique elementφu ofH*, defined bywhere moreover,
The Riesz representation theorem states that the map fromH toH* defined byu ↦φu issurjective, which makes this map anisometricantilinear isomorphism.[75] So to every elementφ of the dualH* there exists one and only oneuφ inH such thatfor allx ∈H. The inner product on the dual spaceH* satisfies
The reversal of order on the right-hand side restores linearity inφ from the antilinearity ofuφ. In the real case, the antilinear isomorphism fromH to its dual is actually an isomorphism, and so real Hilbert spaces are naturally isomorphic to their own duals.
The representing vectoruφ is obtained in the following way. Whenφ ≠ 0, thekernelF = Ker(φ) is a closed vector subspace ofH, not equal toH, hence there exists a nonzero vectorv orthogonal toF. The vectoru is a suitable scalar multipleλv ofv. The requirement thatφ(v) = ⟨v,u⟩ yields
This correspondenceφ ↔u is exploited by thebra–ket notation popular inphysics.[76] It is common in physics to assume that the inner product, denoted by⟨x|y⟩, is linear on the right,The result⟨x|y⟩ can be seen as the action of the linear functional⟨x| (thebra) on the vector|y⟩ (theket).
The Riesz representation theorem relies fundamentally not just on the presence of an inner product, but also on the completeness of the space. In fact, the theorem implies that thetopological dual of any inner product space can be identified with its completion.[77] An immediate consequence of the Riesz representation theorem is also that a Hilbert spaceH isreflexive, meaning that the natural map fromH into itsdouble dual space is an isomorphism.
Conversely, every bounded sequence in a Hilbert space admits weakly convergent subsequences (Alaoglu's theorem).[78] This fact may be used to prove minimization results for continuousconvex functionals, in the same way that theBolzano–Weierstrass theorem is used for continuous functions onRd. Among several variants, one simple statement is as follows:[79]
Iff :H →R is a convex continuous function such thatf(x) tends to+∞ when‖x‖ tends to∞, thenf admits a minimum at some pointx0 ∈H.
This fact (and its various generalizations) are fundamental fordirect methods in thecalculus of variations. Minimization results for convex functionals are also a direct consequence of the slightly more abstract fact that closed bounded convex subsets in a Hilbert spaceH areweakly compact, sinceH is reflexive. The existence of weakly convergent subsequences is a special case of theEberlein–Šmulian theorem.
Any general property ofBanach spaces continues to hold for Hilbert spaces. Theopen mapping theorem states that acontinuoussurjective linear transformation from one Banach space to another is anopen mapping meaning that it sends open sets to open sets. A corollary is thebounded inverse theorem, that a continuous andbijective linear function from one Banach space to another is an isomorphism (that is, a continuous linear map whose inverse is also continuous). This theorem is considerably simpler to prove in the case of Hilbert spaces than in general Banach spaces.[80] The open mapping theorem is equivalent to theclosed graph theorem, which asserts that a linear function from one Banach space to another is continuous if and only if its graph is aclosed set.[81] In the case of Hilbert spaces, this is basic in the study ofunbounded operators (seeClosed operator).
The (geometrical)Hahn–Banach theorem asserts that a closed convex set can be separated from any point outside it by means of ahyperplane of the Hilbert space. This is an immediate consequence of thebest approximation property: ify is the element of a closed convex setF closest tox, then the separating hyperplane is the plane perpendicular to the segmentxy passing through its midpoint.[82]
The sum and the composite of two bounded linear operators is again bounded and linear. Fory inH2, the map that sendsx ∈H1 to⟨Ax,y⟩ is linear and continuous, and according to theRiesz representation theorem can therefore be represented in the formfor some vectorA*y inH1. This defines another bounded linear operatorA* :H2 →H1, theadjoint ofA. The adjoint satisfiesA** =A. When the Riesz representation theorem is used to identify each Hilbert space with its continuous dual space, the adjoint ofA can be shown to beidentical to thetransposetA :H2* →H1* ofA, which by definition sends to the functional
The setB(H) of all bounded linear operators onH (meaning operatorsH →H), together with the addition and composition operations, the norm and the adjoint operation, is aC*-algebra, which is a type ofoperator algebra.
An elementA ofB(H) is called 'self-adjoint' or 'Hermitian' ifA* =A. IfA is Hermitian and⟨Ax,x⟩ ≥ 0 for everyx, thenA is called 'nonnegative', writtenA ≥ 0; if equality holds only whenx = 0, thenA is called 'positive'. The set of self adjoint operators admits apartial order, in whichA ≥B ifA −B ≥ 0. IfA has the formB*B for someB, thenA is nonnegative; ifB is invertible, thenA is positive. A converse is also true in the sense that, for a non-negative operatorA, there exists a unique non-negativesquare rootB such that
In a sense made precise by thespectral theorem, self-adjoint operators can usefully be thought of as operators that are "real". An elementA ofB(H) is callednormal ifA*A =AA*. Normal operators decompose into the sum of a self-adjoint operator and an imaginary multiple of a self adjoint operatorthat commute with each other. Normal operators can also usefully be thought of in terms of their real and imaginary parts.
An elementU ofB(H) is calledunitary ifU is invertible and its inverse is given byU*. This can also be expressed by requiring thatU be onto and⟨Ux,Uy⟩ = ⟨x,y⟩ for allx,y ∈H. The unitary operators form agroup under composition, which is theisometry group ofH.
An element ofB(H) iscompact if it sends bounded sets torelatively compact sets. Equivalently, a bounded operatorT is compact if, for any bounded sequence{xk}, the sequence{Txk} has a convergent subsequence. Manyintegral operators are compact, and in fact define a special class of operators known asHilbert–Schmidt operators that are especially important in the study ofintegral equations.Fredholm operators differ from a compact operator by a multiple of the identity, and are equivalently characterized as operators with a finite dimensionalkernel andcokernel. The index of a Fredholm operatorT is defined by
Unbounded operators are also tractable in Hilbert spaces, and have important applications toquantum mechanics.[84] An unbounded operatorT on a Hilbert spaceH is defined as a linear operator whose domainD(T) is a linear subspace ofH. Often the domainD(T) is a dense subspace ofH, in which caseT is known as adensely defined operator.
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 theobservables in the mathematical formulation of quantum mechanics. Examples of self-adjoint unbounded operators on the Hilbert spaceL2(R) are:[85]
A suitable extension of the differential operator wherei is the imaginary unit andf is a differentiable function of compact support.
The multiplication-by-x operator:
These correspond to themomentum andposition observables, respectively. NeitherA norB is defined on all ofH, since in the case ofA the derivative need not exist, and in the case ofB the product function need not be square integrable. In both cases, the set of possible arguments form dense subspaces ofL2(R).
Two Hilbert spacesH1 andH2 can be combined into another Hilbert space, called the(orthogonal) direct sum,[86] and denoted
consisting of the set of allordered pairs(x1,x2) wherexi ∈Hi,i = 1, 2, and inner product defined by
More generally, ifHi is a family of Hilbert spaces indexed byi ∈I, then the direct sum of theHi, denotedconsists of the set of all indexed familiesin theCartesian product of theHi such that
The inner product is defined by
Each of theHi is included as a closed subspace in the direct sum of all of theHi. Moreover, theHi are pairwise orthogonal. Conversely, if there is a system of closed subspaces,Vi,i ∈I, in a Hilbert spaceH, that are pairwise orthogonal and whose union is dense inH, thenH is canonically isomorphic to the direct sum ofVi. In this case,H is called the internal direct sum of theVi. A direct sum (internal or external) is also equipped with a family of orthogonal projectionsEi onto theith direct summandHi. These projections are bounded, self-adjoint,idempotent operators that satisfy the orthogonality condition
Thespectral theorem forcompact self-adjoint operators on a Hilbert spaceH states thatH 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 theFock space of a system containing a variable number of particles, where each Hilbert space in the direct sum corresponds to an additionaldegree of freedom for the quantum mechanical system. Inrepresentation theory, thePeter–Weyl theorem guarantees that anyunitary representation of acompact group on a Hilbert space splits as the direct sum of finite-dimensional representations.
Ifx1,y1 ∊H1 andx2,y2 ∊H2, then one defines an inner product on the (ordinary)tensor product as follows. Onsimple tensors, let
This formula then extends bysesquilinearity to an inner product onH1 ⊗H2. The Hilbertian tensor product ofH1 andH2, sometimes denoted byH1H2, is the Hilbert space obtained by completingH1 ⊗H2 for the metric associated to this inner product.[87]
An example is provided by the Hilbert spaceL2([0, 1]). The Hilbertian tensor product of two copies ofL2([0, 1]) is isometrically and linearly isomorphic to the spaceL2([0, 1]2) of square-integrable functions on the square[0, 1]2. This isomorphism sends a simple tensorf1 ⊗f2 to the functionon the square.
This example is typical in the following sense.[88] Associated to every simple tensor productx1 ⊗x2 is the rank one operator fromH∗ 1 toH2 that maps a givenx* ∈H∗ 1 as
This mapping defined on simple tensors extends to a linear identification betweenH1 ⊗H2 and the space of finite rank operators fromH∗ 1 toH2. This extends to a linear isometry of the Hilbertian tensor productH1H2 with the Hilbert spaceHS(H∗ 1,H2) ofHilbert–Schmidt operators fromH∗ 1 toH2.
The notion of anorthonormal basis from linear algebra generalizes over to the case of Hilbert spaces.[89] In a Hilbert spaceH, an orthonormal basis is a family{ek}k ∈B of elements ofH satisfying the conditions:
Orthogonality: Every two different elements ofB are orthogonal:⟨ek,ej⟩ = 0 for allk,j ∈B withk ≠j.
Normalization: Every element of the family has norm 1:‖ek‖ = 1 for allk ∈B.
A system of vectors satisfying the first two conditions basis is called an orthonormal system or an orthonormal set (or an orthonormal sequence ifB iscountable). Such a system is alwayslinearly independent.
Despite the name, an orthonormal basis is not, in general, a basis in the sense of linear algebra (Hamel basis). More precisely, an orthonormal basis is a Hamel basis if and only if the Hilbert space is a finite-dimensional vector space.[90]
Completeness of an orthonormal system of vectors of a Hilbert space can be equivalently restated as:
for everyv ∈H, if⟨v,ek⟩ = 0 for allk ∈B, thenv =0.
This is related to the fact that the only vector orthogonal to a dense linear subspace is the zero vector, for ifS is any orthonormal set andv is orthogonal toS, thenv is orthogonal to the closure of the linear span ofS, which is the whole space.
Examples of orthonormal bases include:
the set{(1, 0, 0), (0, 1, 0), (0, 0, 1)} forms an orthonormal basis ofR3 with thedot product;
the sequence{ fn |n ∈Z} withfn(x) =exp(2πinx) forms an orthonormal basis of the complex spaceL2([0, 1]);
In the infinite-dimensional case, an orthonormal basis will not be a basis in the sense oflinear algebra; to distinguish the two, the latter basis is also called aHamel basis. That the span of the basis vectors is dense implies that every vector in the space can be written as the sum of an infinite series, and the orthogonality implies that this decomposition is unique.
The space of square-summable sequences of complex numbers is the set of infinite sequences[9]of real or complex numbers such that
This space has an orthonormal basis:
This space is the infinite-dimensional generalization of the space of finite-dimensional vectors. It is usually the first example used to show that in infinite-dimensional spaces, a set that isclosed andbounded is not necessarily(sequentially) compact (as is the case in allfinite dimensional spaces). Indeed, the set of orthonormal vectors above shows this: It is an infinite sequence of vectors in the unit ball (i.e., the ball of points with norm less than or equal one). This set is clearly bounded and closed; yet, no subsequence of these vectors converges to anything and consequently the unit ball in is not compact. Intuitively, this is because "there is always another coordinate direction" into which the next elements of the sequence can evade.
One can generalize the space in many ways. For example, ifB is any set, then one can form a Hilbert space of sequences with index setB, defined by[91]
The summation overB is here defined bythesupremum being taken over all finite subsets of B. It follows that, for this sum to be finite, every element ofl2(B) has only countably many nonzero terms. This space becomes a Hilbert space with the inner product
for allx,y ∈l2(B). Here the sum also has only countably many nonzero terms, and is unconditionally convergent by the Cauchy–Schwarz inequality.
An orthonormal basis ofl2(B) is indexed by the setB, given by
Letf1, ...,fn be a finite orthonormal system in H. For an arbitrary vectorx ∈H, let
Then⟨x,fk⟩ = ⟨y,fk⟩ for everyk = 1, ...,n. It follows thatx −y is orthogonal to eachfk, hencex −y is orthogonal to y. Using the Pythagorean identity twice, it follows that
Let{fi},i ∈I, be an arbitrary orthonormal system in H. Applying the preceding inequality to every finite subsetJ ofI gives Bessel's inequality:[92](according to the definition of thesum of an arbitrary family of non-negative real numbers).
Geometrically, Bessel's inequality implies that the orthogonal projection ofx onto the linear subspace spanned by thefi has norm that does not exceed that ofx. In two dimensions, this is the assertion that the length of the leg of a right triangle may not exceed the length of the hypotenuse.
Bessel's inequality is a stepping stone to the stronger result calledParseval's identity, which governs the case when Bessel's inequality is actually an equality. By definition, if{ek}k ∈B is an orthonormal basis ofH, then every elementx ofH may be written as
Even ifB is uncountable, Bessel's inequality guarantees that the expression is well-defined and consists only of countably many nonzero terms. This sum is called the Fourier expansion ofx, and the individual coefficients⟨x,ek⟩ are the Fourier coefficients ofx. Parseval's identity then asserts that[93]
Conversely,[93] if{ek} is an orthonormal set such that Parseval's identity holds for everyx, then{ek} is an orthonormal basis.
As a consequence ofZorn's lemma,every Hilbert space admits an orthonormal basis; furthermore, any two orthonormal bases of the same space have the samecardinality, called the Hilbert dimension of the space.[94] For instance, sincel2(B) has an orthonormal basis indexed byB, its Hilbert dimension is the cardinality ofB (which may be a finite integer, or a countable or uncountablecardinal number).
The Hilbert dimension is not greater than theHamel dimension (the usual dimension of a vector space).
As a consequence of Parseval's identity,[95] if{ek}k ∈B is an orthonormal basis ofH, then the mapΦ :H →l2(B) defined byΦ(x) = ⟨x,ek⟩k∈B is an isometric isomorphism of Hilbert spaces: it is a bijective linear mapping such thatfor allx,y ∈H. Thecardinal number ofB is the Hilbert dimension ofH. Thus every Hilbert space is isometrically isomorphic to a sequence spacel2(B) for some setB.
By definition, a Hilbert space isseparable provided it contains a dense countable subset. Along with Zorn's lemma, this means a Hilbert space is separable if and only if it admits acountable orthonormal basis. All infinite-dimensional separable Hilbert spaces are therefore isometrically isomorphic to thesquare-summable sequence space
In the past, Hilbert spaces were often required to be separable as part of the definition.[96]
Most spaces used in physics are separable, and since these are all isomorphic to each other, one often refers to any infinite-dimensional separable Hilbert space as "the Hilbert space" or just "Hilbert space".[97] Even inquantum field theory, most of the Hilbert spaces are in fact separable, as stipulated by theWightman axioms. However, it is sometimes argued that non-separable Hilbert spaces are also important in quantum field theory, roughly because the systems in the theory possess an infinite number ofdegrees of freedom and any infiniteHilbert tensor product (of spaces of dimension greater than one) is non-separable.[98] For instance, abosonic field can be naturally thought of as an element of a tensor product whose factors represent harmonic oscillators at each point of space. From this perspective, the natural state space of a boson might seem to be a non-separable space.[98] However, it is only a small separable subspace of the full tensor product that can contain physically meaningful fields (on which the observables can be defined). Another non-separable Hilbert space models the state of an infinite collection of particles in an unbounded region of space. An orthonormal basis of the space is indexed by the density of the particles, a continuous parameter, and since the set of possible densities is uncountable, the basis is not countable.[98]
IfS is a subset of a Hilbert spaceH, the set of vectors orthogonal toS is defined by
The setS⊥ is aclosed subspace ofH (can be proved easily using the linearity and continuity of the inner product) and so forms itself a Hilbert space. IfV is a closed subspace ofH, thenV⊥ is called theorthogonal complement ofV. In fact, everyx ∈H can then be written uniquely asx =v +w, withv ∈V andw ∈V⊥. Therefore,H is the internal Hilbert direct sum ofV andV⊥.
The linear operatorPV :H →H that mapsx tov is called theorthogonal projection ontoV. There is anatural one-to-one correspondence between the set of all closed subspaces ofH and the set of all bounded self-adjoint operatorsP such thatP2 =P. Specifically,
Theorem—The orthogonal projectionPV is a self-adjoint linear operator onH of norm ≤ 1 with the propertyP2 V =PV. Moreover, any self-adjoint linear operatorE such thatE2 =E is of the formPV, whereV is the range ofE. For everyx inH,PV(x) is the unique elementv ofV thatminimizes the distance‖x −v‖.
This provides the geometrical interpretation ofPV(x): it is the best approximation tox by elements ofV.[99]
ProjectionsPU andPV are called mutually orthogonal ifPUPV = 0. This is equivalent toU andV being orthogonal as subspaces ofH. The sum of the two projectionsPU andPV is a projection only ifU andV are orthogonal to each other, and in that casePU +PV =PU+V.[100] The compositePUPV is generally not a projection; in fact, the composite is a projection if and only if the two projections commute, and in that casePUPV =PU∩V.[101]
By restricting the codomain to the Hilbert spaceV, the orthogonal projectionPV gives rise to a projection mappingπ :H →V; it is the adjoint of theinclusion mappingmeaning thatfor allx ∈V andy ∈H.
The operator norm of the orthogonal projectionPV onto a nonzero closed subspaceV is equal to 1:
Every closed subspaceV of a Hilbert space is therefore the image of an operatorP of norm one such thatP2 =P. The property of possessing appropriate projection operators characterizes Hilbert spaces:[102]
A Banach space of dimension higher than 2 is (isometrically) a Hilbert space if and only if, for every closed subspaceV, there is an operatorPV of norm one whose image isV such thatP2 V =PV.
While this result characterizes the metric structure of a Hilbert space, the structure of a Hilbert space as atopological vector space can itself be characterized in terms of the presence of complementary subspaces:[103]
A Banach spaceX is topologically and linearly isomorphic to a Hilbert space if and only if, to every closed subspaceV, there is a closed subspaceW such thatX is equal to the internal direct sumV ⊕W.
The orthogonal complement satisfies some more elementary results. It is amonotone function in the sense that ifU ⊂V, thenV⊥ ⊆U⊥ with equality holding if and only ifV is contained in theclosure ofU. This result is a special case of theHahn–Banach theorem. The closure of a subspace can be completely characterized in terms of the orthogonal complement: ifV is a subspace ofH, then the closure ofV is equal toV⊥⊥. The orthogonal complement is thus aGalois connection on thepartial order of subspaces of a Hilbert space. In general, the orthogonal complement of a sum of subspaces is the intersection of the orthogonal complements:[104]
There is a well-developedspectral theory for self-adjoint operators in a Hilbert space, that is roughly analogous to the study ofsymmetric matrices over the reals or self-adjoint matrices over the complex numbers.[105] In the same sense, one can obtain a "diagonalization" of a self-adjoint operator as a suitable sum (actually an integral) of orthogonal projection operators.
Thespectrum of an operatorT, denotedσ(T), is the set of complex numbersλ such thatT −λ lacks a continuous inverse. IfT is bounded, then the spectrum is always acompact set in the complex plane, and lies inside the disc|z| ≤ ‖T‖. IfT is self-adjoint, then the spectrum is real. In fact, it is contained in the interval[m,M] where
Moreover,m andM are both actually contained within the spectrum.
The eigenspaces of an operatorT are given by
Unlike with finite matrices, not every element of the spectrum ofT must be an eigenvalue: the linear operatorT −λ may only lack an inverse because it is not surjective. Elements of the spectrum of an operator in the general sense are known asspectral values. Since spectral values need not be eigenvalues, the spectral decomposition is often more subtle than in finite dimensions.
A compact self-adjoint operatorT has only countably (or finitely) many spectral values. The spectrum ofT has nolimit point in the complex plane except possibly zero. The eigenspaces ofT decomposeH into an orthogonal direct sum: Moreover, ifEλ denotes the orthogonal projection onto the eigenspaceHλ, then where the sum converges with respect to the norm onB(H).
This theorem plays a fundamental role in the theory ofintegral equations, as many integral operators are compact, in particular those that arise fromHilbert–Schmidt operators.
The general spectral theorem for self-adjoint operators involves a kind of operator-valuedRiemann–Stieltjes integral, rather than an infinite summation.[107] Thespectral family associated toT associates to each real number λ an operatorEλ, which is the projection onto the nullspace of the operator(T −λ)+, where the positive part of a self-adjoint operator is defined by
The operatorsEλ are monotone increasing relative to the partial order defined on self-adjoint operators; the eigenvalues correspond precisely to the jump discontinuities. One has the spectral theorem, which asserts
The integral is understood as a Riemann–Stieltjes integral, convergent with respect to the norm onB(H). In particular, one has the ordinary scalar-valued integral representation
A somewhat similar spectral decomposition holds for normal operators, although because the spectrum may now contain non-real complex numbers, the operator-valued Stieltjes measuredEλ must instead be replaced by aresolution of the identity.
A major application of spectral methods is thespectral mapping theorem, which allows one to apply to a self-adjoint operatorT any continuous complex functionf defined on the spectrum ofT by forming the integral
The spectral theory ofunbounded self-adjoint operators is only marginally more difficult than for bounded operators. The spectrum of an unbounded operator is defined in precisely the same way as for bounded operators:λ is a spectral value if theresolvent operator
fails to be a well-defined continuous operator. The self-adjointness ofT still guarantees that the spectrum is real. Thus the essential idea of working with unbounded operators is to look instead at the resolventRλ whereλ is nonreal. This is abounded normal operator, which admits a spectral representation that can then be transferred to a spectral representation ofT itself. A similar strategy is used, for instance, to study the spectrum of the Laplace operator: rather than address the operator directly, one instead looks as an associated resolvent such as aRiesz potential orBessel potential.
A precise version of the spectral theorem in this case is:[109]
Theorem—Given a densely defined self-adjoint operatorT on a Hilbert spaceH, there corresponds a uniqueresolution of the identityE on the Borel sets ofR, such that for allx ∈D(T) andy ∈H. The spectral measureE is concentrated on the spectrum ofT.
There is also a version of the spectral theorem that applies to unbounded normal operators.
^However, some sources call finite-dimensional spaces with these properties pre-Hilbert spaces, reserving the term "Hilbert space" for infinite-dimensional spaces; see, e.g.,Levitan 2001.
^InDunford & Schwartz (1958, §IV.16), the result that every linear functional onL2[0,1] is represented by integration is jointly attributed toFréchet (1907) andRiesz (1907). The general result, that the dual of a Hilbert space is identified with the Hilbert space itself, can be found inRiesz (1934).
^Hermann Weyl (2009), "Mind and nature",Mind and nature: selected writings on philosophy, mathematics, and physics, Princeton University Press
^Berthier, M. (2020), "Geometry of color perception. Part 2: perceived colors from real quantum states and Hering's rebit",The Journal of Mathematical Neuroscience,10 (1): 14,doi:10.1186/s13408-020-00092-x,PMC7481323,PMID32902776
^Blanchet, Gérard; Charbit, Maurice (2014).Digital Signal and Image Processing Using MATLAB. Vol. 1 (Second ed.). New Jersey: Wiley. pp. 349–360.ISBN978-1848216402.
^Levitan 2001. Many authors, such asDunford & Schwartz (1958, §IV.4), refer to this just as the dimension. Unless the Hilbert space is finite dimensional, this is not the same thing as its dimension as a linear space (the cardinality of a Hamel basis).
^von Neumann (1955) defines a Hilbert space via a countable Hilbert basis, which amounts to an isometric isomorphism withl2. The convention still persists in most rigorous treatments of quantum mechanics; see for instanceSobrino 1996, Appendix B.
Buttazzo, Giuseppe; Giaquinta, Mariano; Hildebrandt, Stefan (1998),One-dimensional variational problems, Oxford Lecture Series in Mathematics and its Applications, vol. 15, The Clarendon Press Oxford University Press,ISBN978-0-19-850465-8,MR1694383.
Clarkson, J. A. (1936), "Uniformly convex spaces",Trans. Amer. Math. Soc.,40 (3):396–414,doi:10.2307/1989630,JSTOR1989630.
Folland, Gerald B. (1989),Harmonic analysis in phase space, Annals of Mathematics Studies, vol. 122, Princeton University Press,ISBN978-0-691-08527-2.
Fréchet, Maurice (1907), "Sur les ensembles de fonctions et les opérations linéaires",C. R. Acad. Sci. Paris,144:1414–1416.
Fréchet, Maurice (1904), "Sur les opérations linéaires",Transactions of the American Mathematical Society,5 (4):493–499,doi:10.2307/1986278,JSTOR1986278.
Giusti, Enrico (2003),Direct Methods in the Calculus of Variations, World Scientific,ISBN978-981-238-043-2.
Kadison, Richard V.; Ringrose, John R. (1983),Fundamentals of the Theory of Operator Algebras, Vol. I: Elementary Theory, New York: Academic Press, Inc.
Karatzas, Ioannis; Shreve, Steven (2019),Brownian Motion and Stochastic Calculus (2nd ed.), Springer,ISBN978-0-387-97655-6
Saks, Stanisław (2005),Theory of the integral (2nd Dover ed.), Dover,ISBN978-0-486-44648-6; originally publishedMonografje Matematyczne, vol. 7, Warszawa, 1937.
