Open access peer-reviewed chapter

A Survey on Hilbert Spaces and Reproducing Kernels

Written By

Baver Okutmuştur

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

DOI: 10.5772/intechopen.91479

From the Edited Volume

Functional Calculus

Edited by Kamal Shah and Baver Okutmuştur

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Abstract

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.

Keywords

  • 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.

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2. Introduction to Hilbert spaces

We start by the definition of a vector space and related topics. Let C be 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 H over the field C, then it follows that

  1. if x,y,zH, then

    x+y=y+xH,x+y+z=x+y+zH;

  2. if k is scalar, then kxH.

Inner product. Let H be a linear space over the complex field C. An inner product on H is a two variable function

:H×HC,satisfying

  1. fg=gf¯forf,gH.

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

  3. ff0forfHandff=0f=0.

Pre-Hilbert space. A pre-Hilbert spaceH is a linear space over the complex field C with an inner product defined on it.

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

f=ff1/2orfH=ffH1/2

where fH and =H denote 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

fgfg.E1

In case if f and g are linearly dependent, then the inequality becomes equality.

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

f+gf+g.E2

In case if f and g are linearly dependent, then the inequality becomes equality.

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

fg=14f+g2fg2+if+ig2fig2forf,gH.E3

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

f+g2+fg2=2f2+2g2.E4

Metric. A metric on a set X is a function d:X×XR satisfying the properties.

  • dxy0 and dxy=0 only if x=y;

  • dxy=dyx;

  • dxydxz+dzy;

for all x,y,zX. Moreover the space Xd is 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,hH and for all λC, where d satisfies all requirements to be a metric, we have

  • dfg0 and equality occurs only if f=g.

  • dfg=dgf.

  • dfgdfh+dhg.

  • dfhgh=dfg.

  • dλfλg=λdfg.

Note. The binary function d given in the metric definition above represents the metric topology in H which is called strong topology or norm topology. As a result, a sequence fn0 in the pre-Hilbert space H converges strongly to f if the condition

fnf0whenevern

is satisfied.

2.2 Introduction to linear operators

Linear operator. A map L from a linear space to another linear space is called linear operator if

Lαf+βg=αLf+βLg

is satisfied for all α,βC and for all f,gH.

Continuous operator. An operator L is 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 L is a linear operator from H to G where H and G are pre-Hilbert spaces, then the Lipschitz constant for L is its norm L and it is defined by

L=supLfG/fH:0fH.E5

Theorem 1. Let L be a linear operator from the pre-Hilbert spaces H to G. Then the followings are mutually equivalent:

  1. L is continuous.

  2. L is bounded, that is,

    supLfG:fHk<

    for 0k<.

  3. L is Lipschitz continuous, that is,

    LfLgGλfgH,

    where 0λ< and f,gH.

Some properties of linear operators. Let BHG be the collection of all continuous linear operators from the pre-Hilbert spaces H to G. Then

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

αL+βMf=αLf+βMf,

where L and M are linear operators, fH and α,βC.

  • Whenever H=G, then BHG is denoted by BH.

  • If K is another pre-Hilbert space, LBHG and KBGK. Then the product

KLf=KLfforfHBHK.

In addition,

  1. KξL+ζM=ξKL+ζKM

  2. ξL=ξL

  3. L+ML+M and

  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 H to the scalar field C is called a linear form (or linear functional).

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

fnfm0whenevern,m,

then there is fH such that

fnf0whenevern.

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 H is embedded as a dense subspace of a Hilbert space H˜ so that

fgH˜=fgHforf,gH

is called completion.

Note. If L is a continuous linear operator from a dense subspace M of a Hilbert space H to a Hilbert space G, then it can be extended uniquely to a continuous linear operator from H to G with preserving norm.

Theorem 2. Let M and N be dense subspaces of the Hilbert spaces H and G, respectively. For fH,gM and 0λ<, if a linear operator L from M to G satisfies

LfgGλfHgG,E6

then L is uniquely extended to a continuous linear operator from M to G with norm λ where the norm coincides with the minimum of such λ.

Theorem 3. Let Ωμ denotes a measure space so that Ω is the union of subsets of finite positive measure and L2Ωμ consists of all measurable functions fω on Ω such that

Ωfω2ω<.E7

Then L2Ωμ is a Hilbert space with respect to the inner product

fgΩfωgω¯ω.E8

Theorem 4 (F. Riesz). For each continuous linear functional φ on a Hilbert space H, there exists uniquely gH such that

φf=fgforfH.E9

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

H=MM

is satisfied. In other words, fH can be uniquely written by

f=fM+fMwithfMM,fMM.E10

In addition, fM coincides with the distance from f to M

fM=minfg:gM.E11

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

Total subset of a Hilbert space. A subset A of a Hilbert space H is called total in H if 0 is the only element that is orthogonal to all elements of A. In other words,

A=0.

As a result, A is total if and only if every element of H can be approximated by linear combinations of elements of A.

Orthogonal projection. If M is a closed subspace of H, the map ffM gives a linear operator from H to M with norm 1. We call this operator as the orthogonal projection to M and denote it by PM.

Note. If I is the identity operator on H, then IPM denotes the orthogonal projection to M, and the relation

f2=PMf2+IPMf2E12

is satisfied for all fH.

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

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

UfAϵ=h:fghg<ϵforgA,

where A is a finite subset of H and ϵ>0. Then a directed net fλ converges weakly to f if and only if

fλgλfgforallgH.

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

LLfgforfH,gG

is called the operator weak topology in the space BHG of continuous linear operators from H to G. In addition, a directed net Lλ converges weakly to L if

LλfgλLfg.

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

LLfforfH

is called the operator strong topology. Moreover a directed net Lλ converges strongly to L if

LλfLfλ0forallfH.

Theorem 6. Let H and G be Hilbert spaces and BHG be a continuous linear operator from H to G. Then

  • the closed unit ball Uf:f1 of H is weakly compact;

  • the closed unit ball L:L1 of BHG is weakly compact.

Theorem 7. Let H be a Hilbert space and AH. Then if A is weakly bounded in the sense

supfAfg<forgH,E13

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

Theorem 8. If H and G are Hilbert spaces and L is a linear operator from H to G, then the strong continuity and weak continuity for L are equivalent.

Theorem 9. Let H and G be Hilbert spaces. Then the following statements for LBHG are mutually equivalent:

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

    supLLLfg<

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

    supLLLf<.

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

    supLLL<.

Theorem 10. A linear operator L from the Hilbert spaces H to G is said to be closed if its graph

GLfLf:fHE14

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

fnf0inHandLfng0inGg=Lf.

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

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

iΦαf+βhg=αΦfg+βΦhgE15
iiΦfαg+βk=α¯Φfg+β¯ΦfkE16

are satisfied where H and G are Hilbert spaces.

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

Φfg=LfgGE17

is bounded in the sense that

ΦfgλfHgGforfH,gG,E18
whereλL.

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

gΦfg¯

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

fGλfHandΦfg=fgGforgG.

Hence ff becomes linear, and as a result we obtain

Φfg=fgG=LfgG.

Adjoint operator. If LBHG, then the unique operator LBGH satisfying

Φfg=fLgHforfH,gGE19

is called the adjoint of L.

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

LfgG=fLgHforfH,gG.E20

Isometric property. The adjoint operation is isometric if

L=Lissatisfied.E21

Remark. Let H,G, and K be Hilbert spaces and KBGK and LBHG be given. Then

KLBHKandKL=LKE22
KerL=RanLandKerL=ClosRanLE23

where KerL is the kernel of L and RanL is the range of L.

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

  1. RanMRanL.

  2. There exists KBH such that M=LK.

  3. There exists 0λ< such that

MgλLgforgG.

Quadric form. Let H be a Hilbert space. A function

φ:HC

is a quadratic form if for all fH and ζC,

φζf=ζ2φfE24

and

φf+g+φfg=2φf+φgE25

are satisfied.

Note. If LBH, the quadratic form φ on H is defined by

φf=LffforfH,E26

and it is bounded

φfλf2forfH,E27

where λL.

Remark. The sesqui-linear form Φ associated with L can be recovered from the quadratic form φ by the equation

Φfg=14φf+gφfg+φf+igφfigE28

for all f,gH.

Self-adjoint operator. A continuous linear operator L on a Hilbert space H is said to be self-adjoint if L=L.

Remark.L is self-adjoint if and only if the associated sesqui-linear form Φ is Hermitian.

Remark. If L is self-adjoint, then the norm of L coincides with the minimum of λ given in (27) for the related quadratic form

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

L=supLff:f1.E29

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

Lff0forallfH.

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

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

Lfg2LffLgg.E30

Theorem 14. Let L and M be continuous positive operators on H and G, respectively. Then a continuous linear operator K from H to G satisfies the inequality

KfgG2LffHMggGforfH,GE31

if and only if the continuous linear operator

LKKM

on the direct sum Hilbert space HG with

fgLf+KgKf+Mg

is positive definite.

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

Modulus operator. The square root of the positive definite operator LL is called the modulus (operator) of L if L is a continuous linear operator.

Isometry. A linear operator U between Hilbert spaces H and G is called isometric or an isometry if

UfG=fHforfHE32

is satisfied, that is, it preserves the norm.

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

UfUgG=fgHforf,gH,E33

that is, U preserves the inner product.

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

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

Partial isometry. A continuous linear operator U between Hilbert spaces H and G is called a partial isometry if

fKerU=RanUUf=f.

The spaces KerU and RanU are called the initial space of U and the final space of U, respectively.

Note. If U is a partial isometry, then its adjoint U is also a partial isometry.

Theorem 17. Every continuous linear operator L on H admits a unique decomposition

L=UL˜,E34

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

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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 H be a Hilbert space of functions on a nonempty set X with the inner product fg and norm f=ff1/2 for fandgH. Then the complex valued function Kyx of y and x in X is called a reproducing kernel ofH if

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

  2. For all xX and all fH,

fx=fKx,E35

are satisfied.

Note. Let K be a reproducing kernel. Applying (35) to the function Kx at y, we get

Kxy=Kyx=KyKx,forx,yX.E36

Then, for any xX, we obtain

Kx=KxKx1/2=Kxx1/2.E37

Note. Observe that the subset KxxX is total in H, that is, its closed linear span coincides with H. This follows from the fact that, if fH and fKx for all xX, then

fx=fKx=0forallxX,

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

RKHS. A Hilbert space H of functions on a set X is called a RKHS if there exists a reproducing kernel K of H.

Theorem 18. If a Hilbert space H of functions on a set X admits a reproducing kernel K, then this reproducing kernel K is unique.

Theorem 19. There exists a reproducing kernel K for H for a Hilbert space H of functions on X, if and only if for all xX, the linear functional Hffx of evaluation at x is bounded on H.

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

i,j=1nϵ¯jϵiKyjyiR.

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

i,j=1nϵ¯jϵiKyjyi0.

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

x,yXϵ¯yϵxKyx0.E38

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

Properties of RKHS. Given a reproducing kernel Hilbert space H and its kernel Kyx on 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=0 for all yX.

    3. fx0=0 for all fH.

    Theorem 21. For any positive definite kernel K on X, there exists a unique Hilbert space HK of functions on X with reproducing kernel K.

    Theorem 22. Every sequence of functions fnn1 that converges strongly to a function f in HKX converges also in the pointwise sense, i.e., for any point xX,

limnfnx=fx.

In addition, this convergence is uniform on every subset of X on which xKxx is bounded.

Theorem 23. A complex valued function g on X belongs to the reproducing kernel Hilbert space HKX if and only if there exists 0λ< such that,

gygx¯λ2KyxonX.E39

g coincides with the minimum of all such λ.

Theorem 24. If K1yx and K2yx are two positive definite kernels on X, then the following statements are mutually equivalent:

  1. HK1XHK2X.

  2. There exists 0λ< such that

K1yxλ2K2yx.

Note. For any map φ from a set X to a Hilbert space H, with the notation xφx, a kernel K can be defined by

Kyx=φxφyforx,yX.E40

Theorem 25. Let φ:XH be an arbitrary map and for x,yX let K be defined as

Kyx=φxφy.

Then K is a positive definite kernel.

Theorem 26. Let T be the linear operator from H to the space of functions on X, defined by

Tfx=fφxforxX,fH.

Then RanT coincides with HKX and

TfK=PMfforfH,

where M is the orthogonal complement of KerT,PM is the orthogonal projection onto M, and K denotes the norm in HKX.

Kolmogorov decomposition. Let Kyx be a positive definite kernel on an abstract set X. Then there exists a Hilbert space H and a function φ:XH such that

Kyx=φxφyforx,yX.

3.2 Operations with RKHSs

Theorem 27. Let K0 be the restriction of the positive definite kernel K to a nonempty subset X0 of X and let HK0X and HKX be the RKHS corresponding to K0 and K, respectively. Then

HK0X0={fX0:fHKX}E41

and

hK0=minfK:fX0=hforallhHK0X0.E42

Remark. If K1yx and K2yx are two positive definite kernels, then

Kyx=K1yx+K2yx

is also a positive definite kernel.

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

HKX=HK1X+HK2X,

and for fHK1X and gHK2X, it follows that

f+gK2=minf+hK12+ghK22:hHK1XHK2X.E43

Theorem 28. The intersection HK1XHK2X of Hilbert spaces HK1X and HK2X is again a Hilbert space of functions on X with respect to the norm

f2fK12+fK22.

In addition the intersection Hilbert space is a RKHS.

Theorem 29. The reproducing kernel of the space

HKX=HK1XHK2X

is determined, as a quadratic form, by

x,yεy¯εxKyx=inf{x,yηy¯ηxK1yx+x,yζy¯ζxK2yx:εx=ηx+ζx},

where ϵx,ηx,ζx are an arbitrary complex valued function on X with finite support.

Theorem 30. The tensor product Hilbert space

HK1XHK2X

is a RKHS on X×X.

Theorem 31. The RKHS HKX of the kernel Kyx=K1yxK2yx consists of all functions f on X for which there are sequences gnn0 of functions in HK1X and hnn0 of functions in HK2X so that

1gnK12hnK22<,1gnxhnx=fx,xX,E44

and the norm is given by

fK2=min1gnK12hnK22,

where the minimum is taken over the set of all sequences gnn0 and hn0 satisfying (44).

3.3 Examples of RKHS. Bergman and Hardy spaces

Bergman space. The space of all analytic functions f on Ω for which

Ωfz2dxdy<,z=x+iy

is satisfied is called the Bergman space on Ω and denoted by A2Ω.

Remark.A2Ω is a RKHS with respect to the inner product

fgfgΩΩfzgz¯dxdy,

and its kernel is called the Bergman kernel on Ω and denoted by BΩwz.

Bergman kernel for the unit disc. The Bergman kernel for the open unit disc D is given by

BDwz=1π11wz¯2forw,zD.E45

Bergman kernel of a simply connected domain. The Bergman kernel of a simply connected domain ΩC is given by

BΩwz=1πφwφz¯1φwφz¯2forw,zΩ,E46

where φ is any conformal mapping function from Ω onto D.

Theorem 32. A conformal mapping from Ω to D can be recovered from the Bergman kernel of Ω.

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

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

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

BΩwz=2π2Gwz¯wz,wz.E47

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

Remark.fL2T belongs to the Hardy space H2T if and only if it is orthonormal to all φn (n<0), that is, all Fourier coefficients of f with negative indices vanish. Then we have

fgL2=n=0anb¯nforf,gH2T,E48

where

an=fφnL2andbn=gφnL2n=01.

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

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Notes

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].

References

  1. 1. Conway JB. A Course in Functional Analysis. Berlin-Heidelberg, New York: Springer-Verlag; 1989
  2. 2. Douglas RG. Banach Algebra Techniques in Operator Theory. New York: Springer-Verlag, Academic Press; 1972
  3. 3. Okutmustur B, Gheondea A. Reproducing Kernel Hilbert Spaces: The Basics, Bergman Spaces, and Interpolation Problems of Reproducing Kernels and its Applications. Riga-Latvia: Lap Lambert Academic Publishing; 2010
  4. 4. Ando T. Reproducing Kernel Spaces and Quadratic Inequalities, Lecture Notes. Sapporo, Japan: Hokkaido University, Research Institute of Applied Electricity, Division of Applied Mathematics; 1987
  5. 5. Aronszajn N. Theory of reproducing kernels. Transactions of the American Mathematical Society. 1950;68:337-404
  6. 6. Saitoh S, Sawano Y. Theory of Reproducing Kernels and Applications. Singapore: Springer; 2016
  7. 7. Duren PL. Theory of Hp Spaces. New York: Academic Press, Inc.; 1970
  8. 8. Duren PL, Schuster A. Bergman Spaces. Providence, R.I.: American Mathematical Society; 2004
  9. 9. Koosis P. Introduction to Hp Spaces. Cambridge: Cambridge Mathematical Press; 1970

Written By

Baver Okutmuştur

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