Open access peer-reviewed chapter

A Survey on Hilbert Spaces and Reproducing Kernels

By Baver Okutmuştur

Submitted: August 1st 2019Reviewed: February 3rd 2020Published: February 28th 2020

DOI: 10.5772/intechopen.91479

Downloaded: 396


The main purpose of this chapter is to provide a brief review of Hilbert space with its fundamental features and introduce reproducing kernels of the corresponding spaces. We separate our analysis into two parts. In the first part, the basic facts on the inner product spaces including the notion of norms, pre-Hilbert spaces, and finally Hilbert spaces are presented. The second part is devoted to the reproducing kernels and the related Hilbert spaces which is called the reproducing kernel Hilbert spaces (RKHS) in the complex plane. The operations on reproducing kernels with some important theorems on the Bergman kernel for different domains are analyzed in this part.


  • Hilbert spaces
  • norm spaces
  • reproducing kernels
  • reproducing kernel Hilbert spaces (RKHS)
  • operations on reproducing kernels
  • sesqui-analytic kernels
  • analytic functions
  • Bergman kernel

1. Framework

This chapter consists of introductory concept on the Hilbert space theory and reproducing kernels. We start by presenting basic definitions, propositions, and theorems from functional analysis related to Hilbert spaces. The notion of linear space, norm, inner product, and pre-Hilbert spaces are in the first part. The second part is devoted to the fundamental properties of the reproducing kernels and the related Hilbert spaces. The operations with reproducing kernels, inclusion property, Bergman kernel, and further properties with examples of the reproducing kernels are analyzed in the latter section.


2. Introduction to Hilbert spaces

We start by the definition of a vector space and related topics. Let Cbe the complex field. The following preliminaries can be considered as fundamental concepts of the Hilbert spaces.

2.1 Vector spaces and inner product spaces

Vector space.A vector space is a linear space that is closed under vector addition and scalar multiplication. More precisely, if we denote our linear space by Hover the field C, then it follows that

  1. if x,y,zH,then


  • if kis scalar, then kxH.

  • Inner product.Let Hbe a linear space over the complex field C. An inner producton His a two variable function


    1. fg=gf¯forf,gH.

    2. αf+βgh=αfh+βghand fαg+βh=α¯fg+β¯fhforα,βCandf,g,hH.

    3. ff0forfHandff=0f=0.

    Pre-Hilbert space.A pre-Hilbert spaceHis a linear space over the complex field Cwith an inner product defined on it.

    Norm space or inner product space.A norm on an inner product space Hdenoted by is defined by


    where fHand =Hdenote the inner product on H. The corresponding space is called as the inner product space or the norm space.

    Properties of norm.For all f,gH, and λC, we have

    • f0. (Observe that the equality occurs only if f=0).

    • λf=λf.

    Schwarz inequality.For all f,gH, it follows that


    In case if fand gare linearly dependent, then the inequality becomes equality.

    Triangle inequality.For all f,gH, it follows that


    In case if fand gare linearly dependent, then the inequality becomes equality.

    Polarization identity.For all f,gH, it follows that


    Parallelogram identity.For all f,gH, it follows that


    Metric.A metric on a set Xis a function d:X×XRsatisfying the properties.

    • dxy0and dxy=0only if x=y;

    • dxy=dyx;

    • dxydxz+dzy;

    for all x,y,zX. Moreover the space Xdis the associated metric space. If we rearrange the metric with its properties for the inner product space H, then it follows that for all f,g,hHand for all λC,where dsatisfies all requirements to be a metric, we have

    • dfg0and equality occurs only if f=g.

    • dfg=dgf.

    • dfgdfh+dhg.

    • dfhgh=dfg.

    • dλfλg=λdfg.

    Note.The binary function dgiven in the metric definition above represents the metric topology in Hwhich is called strong topologyor norm topology.As a result, a sequence fn0in the pre-Hilbert space Hconverges strongly to fif the condition


    is satisfied.

    2.2 Introduction to linear operators

    Linear operator.A map Lfrom a linear space to another linear space is called linear operatorif


    is satisfied for all α,βCand for all f,gH.

    Continuous operator.An operator Lis said to be continuous if it is continuous at each point of its domain. Notice that the domain and range spaces must be convenient for appropriate topologies.

    Lipschitz constant of a linear operator.If Lis a linear operator from Hto Gwhere Hand Gare pre-Hilbert spaces, then the Lipschitz constant for Lis its norm Land it is defined by


    Theorem 1.Let Lbe a linear operator from the pre-Hilbert spaces Hto G.Then the followings are mutually equivalent:

    1. Lis continuous.

    2. Lis bounded, that is,


    for 0k<.

  • Lis Lipschitz continuous, that is,


    where 0λ<and f,gH.

  • Some properties of linear operators.Let BHGbe the collection of all continuous linear operators from the pre-Hilbert spaces Hto G. Then

    • BHGis a linear space with respect to the natural addition and scalar multiplication satisfying


    where Land Mare linear operators, fHand α,βC.

    • Whenever H=G, then BHGis denoted by BH.

    • If Kis another pre-Hilbert space, LBHGand KBGK. Then the product


    In addition,

    1. KξL+ζM=ξKL+ζKM

    2. ξL=ξL

    3. L+ML+Mand

    4. KLKL.

    are also satisfied.

    2.3 Hilbert spaces and linear operators

    Linear form (or linear functional).A linear operator from the pre-Hilbert space Hto the scalar field Cis called a linear form(or linear functional).

    Hilbert spaces.A pre-Hilbert space His said to be a Hilbert spaceif it is complete in metric. In other words if fnis a Cauchy sequence in H, that is, if


    then there is fHsuch that


    Note.Every subspace of a pre-Hilbert space is also a pre-Hilbert space with respect to the induced inner product. However, the reverse is not always true. For a subspace of a Hilbert space to be also a Hilbert space, it must be closed.

    Completion.The canonical method for which a pre-Hilbert space His embedded as a dense subspace of a Hilbert space H˜so that


    is called completion.

    Note.If Lis a continuous linear operator from a dense subspace Mof a Hilbert space Hto a Hilbert space G, then it can be extended uniquely to a continuous linear operator from Hto Gwith preserving norm.

    Theorem 2.Let Mand Nbe dense subspaces of the Hilbert spaces Hand G, respectively. For fH,gMand 0λ<, if a linear operator Lfrom Mto Gsatisfies


    then Lis uniquely extended to a continuous linear operator from Mto Gwith norm λwhere the norm coincides with the minimum of such λ.

    Theorem 3.Let Ωμdenotes a measure space so that Ωis the union of subsets of finite positive measure and L2Ωμconsists of all measurable functions fωon Ωsuch that


    Then L2Ωμis a Hilbert space with respect to the inner product


    Theorem 4 (F. Riesz).For each continuous linear functional φon a Hilbert space H, there exists uniquely gHsuch that


    Theorem 5.Let Mbe a closed subspace of a Hilbert space H.Then the algebraic direct sum relation


    is satisfied. In other words, fHcan be uniquely written by


    In addition, fMcoincides with the distance from fto M


    Remark.In a Hilbert space, the closed linear span of any subset Aof a Hilbert space Hcoincides with A.

    Total subset of a Hilbert space.A subset Aof a Hilbert space His called totalin Hif 0is the only element that is orthogonal to all elements of A. In other words,


    As a result, Ais total if and only if every element of Hcan be approximated by linear combinations of elements of A.

    Orthogonal projection.If Mis a closed subspace of H, the map ffMgives a linear operator from Hto Mwith norm 1. We call this operator as the orthogonal projectionto Mand denote it by PM.

    Note.If Iis the identity operator on H, then IPMdenotes the orthogonal projection to M, and the relation


    is satisfied for all fH.

    Weak topology.The weakest topology that makes continuous all linear functionals of the form ffgis called the weak topologyof a Hilbert space H.

    Note.If fH, then with respect to the weak topology, a fundamental system of neighborhoods of fis composed of subsets of the form


    where Ais a finite subset of Hand ϵ>0. Then a directed net fλconverges weakly to fif and only if


    Operator weak topology.The weakest topology that makes continuous all linear functionals of the form


    is called the operator weak topologyin the space BHGof continuous linear operators from Hto G. In addition, a directed net Lλconverges weakly to Lif


    Operator strong topology.The weakest topology that makes continuous all linear operators of the form


    is called the operator strong topology. Moreover a directed net Lλconverges strongly to Lif


    Theorem 6. Let Hand Gbe Hilbert spaces and BHGbe a continuous linear operator from Hto G. Then

    • the closed unit ball Uf:f1of His weakly compact;

    • the closed unit ball L:L1of BHGis weakly compact.

    Theorem 7.Let Hbe a Hilbert space and AH.Then if Ais weakly bounded in the sense


    then it is strongly bounded, that is, supfAf<.

    Theorem 8.If Hand Gare Hilbert spaces and Lis a linear operator from Hto G, then the strong continuity and weak continuity for Lare equivalent.

    Theorem 9.Let Hand Gbe Hilbert spaces. Then the following statements for LBHGare mutually equivalent:

    1. Lis weakly bounded; that is, for fH,gG, we have


    2. Lis strongly bounded; that is, for fH,we have


    3. Lis norm bounded (or uniformly bounded); that is,


    Theorem 10.A linear operator Lfrom the Hilbert spaces Hto Gis said to be closedif its graph


    is a closed subspace of the direct sum space HG, that is, whenever n,


    Theorem 11.If Lis a closed linear operator with a domain of a Hilbert space Hto another Hilbert space G, then it is continuous.

    Sesqui-linear form.A function Φ:H×GCis a sesqui-linear form(or sesqui-linear function) if for f,hH,g,kGand α,βC,


    are satisfied where Hand Gare Hilbert spaces.

    Remark.If LBHG, then the sesqui-linear form Φdefined by


    is bounded in the sense that


    Remark.If a sesqui-linear form Φsatisfies the condition (18), then for fH,the linear functional


    is continuous on G.If we apply the Riesz theorem, then there exists uniquely fGsatisfying


    Hence ffbecomes linear, and as a result we obtain


    Adjoint operator.If LBHG,then the unique operator LBGHsatisfying


    is called the adjointof L.

    Remark.By the definitions of Land L,it follows that


    Isometric property.The adjoint operation is isometric if


    Remark.Let H,G, and Kbe Hilbert spaces and KBGKand LBHGbe given. Then


    where KerLis the kernel of Land RanLis the range of L.

    Theorem 12.If L,MBHG,then the following statements are mutually equivalent.

    1. RanMRanL.

    2. There exists KBHsuch that M=LK.

    3. There exists 0λ<such that


    Quadric form.Let Hbe a Hilbert space. A function


    is a quadratic formif for all fHand ζC,




    are satisfied.

    Note.If LBH,the quadratic form φon His defined by


    and it is bounded


    where λL.

    Remark.The sesqui-linear form Φassociated with Lcan be recovered from the quadratic form φby the equation


    for all f,gH.

    Self-adjoint operator.A continuous linear operator Lon a Hilbert space His said to be self-adjointif L=L.

    Remark.Lis self-adjoint if and only if the associated sesqui-linear form Φis Hermitian.

    Remark. If Lis self-adjoint, then the norm of Lcoincides with the minimum of λgiven in (27) for the related quadratic form

    Theorem 13.If Lis a continuous self-adjoint operator, then


    Positive definite operator.A self-adjoint operator LBHis said to be positive(or positive definite) if


    If Lff=0only when f=0,then Lis said to be strictly positive(or, strictly positive definite).

    Note.For any positive operator LBH, the Schwarz inequality holds in the following sense


    Theorem 14.Let Land Mbe continuous positive operators on Hand G, respectively. Then a continuous linear operator Kfrom Hto Gsatisfies the inequality


    if and only if the continuous linear operator


    on the direct sum Hilbert space HGwith


    is positive definite.

    Theorem 15.Let Lbe a continuous positive definite operator. Then there exists a unique positive definite operator called the square rootof L,denoted by L1/2,such that L1/22=L.

    Modulus operator.The square root of the positive definite operator LLis called the modulus(operator) of Lif Lis a continuous linear operator.

    Isometry.A linear operator Ubetween Hilbert spaces Hand Gis called isometricor an isometryif


    is satisfied, that is, it preserves the norm.

    Note.Eq. (32) implies that a continuous linear operator Uis isometric if and only if UU=IH;in other words,


    that is, Upreserves the inner product.

    Unitary operator.A surjective isometry linear operator U:HHis called a unitary(operator).

    Note.Observe that if UBHis a unitary operator, then U=U1.

    Partial isometry.A continuous linear operator Ubetween Hilbert spaces Hand Gis called a partial isometryif


    The spaces KerUand RanUare called the initial spaceof Uand the final spaceof U, respectively.

    Note.If Uis a partial isometry, then its adjoint Uis also a partial isometry.

    Theorem 17.Every continuous linear operator Lon Hadmits a unique decomposition


    where L˜is a positive definite operator and Uis a partial isometry with initial space the closure of RanL˜.

    3. Reproducing kernels and RKHS

    We continue our analysis on the abstract theory of reproducing kernels.

    3.1 Definition and fundamental properties

    Reproducing kernels.Let Hbe a Hilbert space of functions on a nonempty set Xwith the inner product fgand norm f=ff1/2for fandgH. Then the complex valued function Kyxof yand xin Xis called a reproducing kernel ofHif

    1. For all xX, it follows that Kx=KxH,

    2. For all xXand all fH,


    are satisfied.

    Note.Let Kbe a reproducing kernel. Applying (35) to the function Kxat y,we get


    Then, for any xX, we obtain


    Note.Observe that the subset KxxXis total in H, that is, its closed linear span coincides with H. This follows from the fact that, if fHand fKxfor all xX, then


    and hence fis the 0element in H. As a result, 0=H.

    RKHS.A Hilbert space Hof functions on a set Xis called a RKHS if there exists a reproducing kernel Kof H.

    Theorem 18.If a Hilbert space Hof functions on a set Xadmits a reproducing kernel K, then this reproducing kernel Kis unique.

    Theorem 19.There exists a reproducing kernel Kfor Hfor a Hilbert space Hof functions on X, if and only if for all xX, the linear functional Hffxof evaluation at xis bounded on H.

    Hermitian and positive definite kernel.Let Xbe an arbitrary set and Kbe a kernel on X, that is, K:X×XC. The kernel Kis called Hermitianif for any finite set of points y1ynX, we have


    It is called positive definite, if for any complex numbers ϵ1,,ϵn, we have


    Note.From the previous inequality, it follows that for any finitely supported family of complex numbers ϵxxX, we have


    Theorem 20.The reproducing kernel Kof a reproducing kernel Hilbert space His a positive definite matrix in the sense of E.H. Moore.

    Properties of RKHS.Given a reproducing kernel Hilbert space Hand its kernel Kyxon X, then for all x,yX, we have

    1. Kyy0.

    2. Kyx=Kxy¯.

    3. Kyx2KyyKxx(Schwarz inequality).

    4. Let x0X. Then the following statements are equivalent:

      1. Kx0x0=0.

      2. Kyx0=0for all yX.

      3. fx0=0for all fH.

      Theorem 21.For any positive definite kernel Kon X, there exists a unique Hilbert space HKof functions on Xwith reproducing kernel K.

      Theorem 22.Every sequence of functions fnn1that converges strongly to a function fin HKXconverges also in the pointwise sense, i.e., for any point xX,


    In addition, this convergence is uniform on every subset of Xon which xKxxis bounded.

    Theorem 23.A complex valued function gon Xbelongs to the reproducing kernel Hilbert space HKXif and only if there exists 0λ<such that,


    gcoincides with the minimum of all such λ.

    Theorem 24.If K1yxand K2yxare two positive definite kernels on X, then the following statements are mutually equivalent:

    1. HK1XHK2X.

    2. There exists 0λ<such that


    Note.For any map φfrom a set Xto a Hilbert space H, with the notation xφx, a kernel Kcan be defined by


    Theorem 25.Let φ:XHbe an arbitrary map and for x,yXlet Kbe defined as


    Then Kis a positive definite kernel.

    Theorem 26.Let Tbe the linear operator from Hto the space of functions on X, defined by


    Then RanTcoincides with HKXand


    where Mis the orthogonal complement of KerT,PMis the orthogonal projection onto M, and Kdenotes the norm in HKX.

    Kolmogorov decomposition.Let Kyxbe a positive definite kernel on an abstract set X. Then there exists a Hilbert space Hand a function φ:XHsuch that


    3.2 Operations with RKHSs

    Theorem 27.Let K0be the restriction of the positive definite kernel Kto a nonempty subset X0of Xand let HK0Xand HKXbe the RKHS corresponding to K0and K, respectively. Then




    Remark.If K1yxand K2yxare two positive definite kernels, then


    is also a positive definite kernel.

    Remark.Let HK1,HK2, and HKbe RKHSs with reproducing kernels K1yx, K2yx, and Kyx, respectively, and let K=K1+K2.Then


    and for fHK1Xand gHK2X,it follows that


    Theorem 28.The intersection HK1XHK2Xof Hilbert spaces HK1Xand HK2Xis again a Hilbert space of functions on Xwith respect to the norm


    In addition the intersection Hilbert space is a RKHS.

    Theorem 29.The reproducing kernel of the space


    is determined, as a quadratic form, by


    where ϵx,ηx,ζxare an arbitrary complex valued function on Xwith finite support.

    Theorem 30.The tensor product Hilbert space


    is a RKHS on X×X.

    Theorem 31.The RKHS HKXof the kernel Kyx=K1yxK2yxconsists of all functions fon Xfor which there are sequences gnn0of functions in HK1Xand hnn0of functions in HK2Xso that


    and the norm is given by


    where the minimum is taken over the set of all sequences gnn0and hn0satisfying (44).

    3.3 Examples of RKHS. Bergman and Hardy spaces

    Bergman space.The space of all analytic functions fon Ωfor which


    is satisfied is called the Bergman spaceon Ωand denoted by A2Ω.

    Remark.A2Ωis a RKHSwith respect to the inner product


    and its kernel is called the Bergman kernelon Ωand denoted by BΩwz.

    Bergman kernel for the unit disc.The Bergman kernel for the open unit disc Dis given by


    Bergman kernel of a simply connected domain.The Bergman kernel of a simply connected domain ΩCis given by


    where φis any conformal mapping function from Ωonto D.

    Theorem 32.A conformal mapping from Ωto Dcan be recovered from the Bergman kernel of Ω.

    Jordan curve.A Jordan curveis a continuous 11image of ξ=1in C.

    Green function.A Green functionGwzof Ωis a function harmonic in Ωexcept at z,where it has logarithmic singularity, and continuous in the closure Ω¯,with boundary values Gwz=0for all w∂Ω,where Ωis a finitely connected domain of the complex plane.

    Theorem 33.Let Ωbe a finitely connected domain bounded by analytic Jordan curves, and let Gwzbe the Green’s function of Ω.Then the Bergman kernel function is


    Hardy space.The closed linear span of φn:n=01in L2(Tis called the (Hilbert type) Hardy spaceon Tand is denoted by H2T.Here φnξ=ξn.

    Remark.fL2Tbelongs to the Hardy space H2Tif and only if it is orthonormal to all φn(n<0), that is, all Fourier coefficients of fwith negative indices vanish. Then we have




    Szegö kernel.The kernel Sξz11ξz¯forξT,zD, or its analytic extension S˜wz11wz¯forw,zDis called the Szegö kernel.


    This chapter intends to offer a sample survey for the fundamental concepts of Hilbert spaces and provide an introductory theory of reproducing kernels. We present the basic properties with important theorems and sometimes with punctual notes and remarks to support the subject. However, due to the limit of content and pages, we skipped the proofs of the theorems. The proofs of the first part can be found in [1, 2] and in most of the basic functional analysis books. Besides, the proofs of the second part (related with the reproducing kernels) can easily be found in [3]. The Hilbert space and functional analysis parts of this chapter are based on the books by J.B. Conway [1] and R.G. Douglas [2]. On the other hand, the reproducing kernel part is based on the lecture notes of T. Ando [4] and N. Aronszajn [5], the book of S. Saitoh and Y. Sawano [6], and the book of B. Okutmustur and A. Gheondea [3]. Moreover, the details of Bergman and Hardy spaces are widely explained in the books [7, 8, 9].

    © 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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    Baver Okutmuştur (February 28th 2020). A Survey on Hilbert Spaces and Reproducing Kernels, Functional Calculus, Kamal Shah and Baver Okutmuştur, IntechOpen, DOI: 10.5772/intechopen.91479. Available from:

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