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
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 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 over the field , then it follows that
if is scalar, then
Inner product. Let be a linear space over the complex field . An inner product on is a two variable function
Pre-Hilbert space. A pre-Hilbert space is a linear space over the complex field with an inner product defined on it.
Norm space or inner product space. A norm on an inner product space denoted by is defined by
where and denote the inner product on . The corresponding space is called as the inner product space or the norm space.
Properties of norm. For all , and , we have
. (Observe that the equality occurs only if ).
Schwarz inequality. For all , it follows that
In case if and are linearly dependent, then the inequality becomes equality.
Triangle inequality. For all , it follows that
In case if and are linearly dependent, then the inequality becomes equality.
Polarization identity. For all , it follows that
Parallelogram identity. For all , it follows that
Metric. A metric on a set is a function satisfying the properties.
and only if ;
for all . Moreover the space is the associated metric space. If we rearrange the metric with its properties for the inner product space , then it follows that for all and for all where satisfies all requirements to be a metric, we have
and equality occurs only if
Note. The binary function given in the metric definition above represents the metric topology in which is called strong topology or norm topology. As a result, a sequence in the pre-Hilbert space converges strongly to if the condition
2.2 Introduction to linear operators
Linear operator. A map from a linear space to another linear space is called linear operator if
is satisfied for all and for all .
Continuous operator. An operator 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 is a linear operator from to where and are pre-Hilbert spaces, then the Lipschitz constant for is its norm and it is defined by
Theorem 1. Let be a linear operator from the pre-Hilbert spaces to Then the followings are mutually equivalent:
is bounded, that is,
is Lipschitz continuous, that is,
Some properties of linear operators. Let be the collection of all continuous linear operators from the pre-Hilbert spaces to . Then
is a linear space with respect to the natural addition and scalar multiplication satisfying
where and are linear operators, and
Whenever , then is denoted by .
If is another pre-Hilbert space, and . Then the product
are also satisfied.
2.3 Hilbert spaces and linear operators
Linear form (or linear functional). A linear operator from the pre-Hilbert space to the scalar field is called a linear form (or linear functional).
Hilbert spaces. A pre-Hilbert space is said to be a Hilbert space if it is complete in metric. In other words if is a Cauchy sequence in , that is, if
then there is such 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 is embedded as a dense subspace of a Hilbert space so that
is called completion.
Note. If is a continuous linear operator from a dense subspace of a Hilbert space to a Hilbert space , then it can be extended uniquely to a continuous linear operator from to with preserving norm.
Theorem 2. Let and be dense subspaces of the Hilbert spaces and , respectively. For and , if a linear operator from to satisfies
then is uniquely extended to a continuous linear operator from to with norm where the norm coincides with the minimum of such .
Theorem 3. Let denotes a measure space so that is the union of subsets of finite positive measure and consists of all measurable functions on such that
Then is a Hilbert space with respect to the inner product
Theorem 4 (F. Riesz). For each continuous linear functional on a Hilbert space , there exists uniquely such that
Theorem 5. Let be a closed subspace of a Hilbert space Then the algebraic direct sum relation
is satisfied. In other words, can be uniquely written by
In addition, coincides with the distance from to
Remark. In a Hilbert space, the closed linear span of any subset of a Hilbert space coincides with
Total subset of a Hilbert space. A subset of a Hilbert space is called total in if is the only element that is orthogonal to all elements of . In other words,
As a result, is total if and only if every element of can be approximated by linear combinations of elements of .
Orthogonal projection. If is a closed subspace of , the map gives a linear operator from to with norm . We call this operator as the orthogonal projection to and denote it by .
Note. If is the identity operator on , then denotes the orthogonal projection to , and the relation
is satisfied for all .
Weak topology. The weakest topology that makes continuous all linear functionals of the form is called the weak topology of a Hilbert space .
Note. If , then with respect to the weak topology, a fundamental system of neighborhoods of f is composed of subsets of the form
where is a finite subset of and . Then a directed net converges weakly to if and only if
Operator weak topology. The weakest topology that makes continuous all linear functionals of the form
is called the operator weak topology in the space of continuous linear operators from to . In addition, a directed net converges weakly to if
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 converges strongly to if
Theorem 6. Let and be Hilbert spaces and be a continuous linear operator from to . Then
the closed unit ball of is weakly compact;
the closed unit ball of is weakly compact.
Theorem 7. Let be a Hilbert space and Then if is weakly bounded in the sense
then it is strongly bounded, that is,
Theorem 8. If and are Hilbert spaces and is a linear operator from to , then the strong continuity and weak continuity for are equivalent.
Theorem 9. Let and be Hilbert spaces. Then the following statements for are mutually equivalent:
is weakly bounded; that is, for , we have
is strongly bounded; that is, for we have
is norm bounded (or uniformly bounded); that is,
Theorem 10. A linear operator from the Hilbert spaces to is said to be closed if its graph
is a closed subspace of the direct sum space , that is, whenever ,
Theorem 11. If is a closed linear operator with a domain of a Hilbert space to another Hilbert space , then it is continuous.
Sesqui-linear form. A function is a sesqui-linear form (or sesqui-linear function) if for and
are satisfied where and are Hilbert spaces.
Remark. If , then the sesqui-linear form defined by
is bounded in the sense that
Remark. If a sesqui-linear form satisfies the condition (18), then for the linear functional
is continuous on If we apply the Riesz theorem, then there exists uniquely satisfying
Hence becomes linear, and as a result we obtain
Adjoint operator. If then the unique operator satisfying
is called the adjoint of .
Remark. By the definitions of and it follows that
Isometric property. The adjoint operation is isometric if
Remark. Let , and be Hilbert spaces and and be given. Then
where is the kernel of and is the range of .
Theorem 12. If then the following statements are mutually equivalent.
There exists such that
There exists such that
Quadric form. Let be a Hilbert space. A function
is a quadratic form if for all and
Note. If the quadratic form on is defined by
and it is bounded
Remark. The sesqui-linear form associated with can be recovered from the quadratic form by the equation
Self-adjoint operator. A continuous linear operator on a Hilbert space is said to be self-adjoint if .
Remark. is self-adjoint if and only if the associated sesqui-linear form is Hermitian.
Remark. If is self-adjoint, then the norm of coincides with the minimum of given in (27) for the related quadratic form
Theorem 13. If is a continuous self-adjoint operator, then
Positive definite operator. A self-adjoint operator is said to be positive (or positive definite) if
If only when then is said to be strictly positive (or, strictly positive definite).
Note. For any positive operator , the Schwarz inequality holds in the following sense
Theorem 14. Let and be continuous positive operators on and , respectively. Then a continuous linear operator from to satisfies the inequality
if and only if the continuous linear operator
on the direct sum Hilbert space with
is positive definite.
Theorem 15. Let be a continuous positive definite operator. Then there exists a unique positive definite operator called the square root of denoted by such that
Modulus operator. The square root of the positive definite operator is called the modulus (operator) of if is a continuous linear operator.
Isometry. A linear operator between Hilbert spaces and is called isometric or an isometry if
is satisfied, that is, it preserves the norm.
Note. Eq. (32) implies that a continuous linear operator is isometric if and only if in other words,
that is, preserves the inner product.
Unitary operator. A surjective isometry linear operator is called a unitary (operator).
Note. Observe that if is a unitary operator, then
Partial isometry. A continuous linear operator between Hilbert spaces and is called a partial isometry if
The spaces and are called the initial space of and the final space of , respectively.
Note. If is a partial isometry, then its adjoint is also a partial isometry.
Theorem 17. Every continuous linear operator on admits a unique decomposition
where is a positive definite operator and is a partial isometry with initial space the closure of .
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 be a Hilbert space of functions on a nonempty set with the inner product and norm for . Then the complex valued function of and in is called a reproducing kernel of if
For all , it follows that ,
For all and all ,
Note. Let be a reproducing kernel. Applying (35) to the function at we get
Then, for any , we obtain
Note. Observe that the subset is total in , that is, its closed linear span coincides with . This follows from the fact that, if and for all , then
and hence is the element in . As a result, .
RKHS. A Hilbert space of functions on a set is called a RKHS if there exists a reproducing kernel of .
Theorem 18. If a Hilbert space of functions on a set admits a reproducing kernel , then this reproducing kernel is unique.
Theorem 19. There exists a reproducing kernel for for a Hilbert space of functions on , if and only if for all , the linear functional of evaluation at is bounded on
Hermitian and positive definite kernel. Let be an arbitrary set and be a kernel on , that is, . The kernel is called Hermitian if for any finite set of points , we have
It is called positive definite, if for any complex numbers , we have
Note. From the previous inequality, it follows that for any finitely supported family of complex numbers , we have
Theorem 20. The reproducing kernel of a reproducing kernel Hilbert space is a positive definite matrix in the sense of E.H. Moore.
Properties of RKHS. Given a reproducing kernel Hilbert space and its kernel on , then for all , we have
Let . Then the following statements are equivalent:
for all .
Theorem 21. For any positive definite kernel on , there exists a unique Hilbert space of functions on with reproducing kernel .
Theorem 22. Every sequence of functions that converges strongly to a function in converges also in the pointwise sense, i.e., for any point ,
In addition, this convergence is uniform on every subset of on which is bounded.
Theorem 23. A complex valued function on belongs to the reproducing kernel Hilbert space if and only if there exists such that,
coincides with the minimum of all such .
Theorem 24. If and are two positive definite kernels on , then the following statements are mutually equivalent:
There exists such that
Note. For any map from a set to a Hilbert space , with the notation , a kernel can be defined by
Theorem 25. Let be an arbitrary map and for let be defined as
Then is a positive definite kernel.
Theorem 26. Let be the linear operator from to the space of functions on , defined by
Then coincides with and
where is the orthogonal complement of is the orthogonal projection onto , and denotes the norm in
Kolmogorov decomposition. Let be a positive definite kernel on an abstract set . Then there exists a Hilbert space and a function such that
3.2 Operations with RKHSs
Theorem 27. Let be the restriction of the positive definite kernel to a nonempty subset of and let and be the RKHS corresponding to and , respectively. Then
Remark. If and are two positive definite kernels, then
is also a positive definite kernel.
Remark. Let , and be RKHSs with reproducing kernels , , and , respectively, and let Then
and for and it follows that
Theorem 28. The intersection of Hilbert spaces and is again a Hilbert space of functions on with 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 are an arbitrary complex valued function on with finite support.
Theorem 30. The tensor product Hilbert space
is a RKHS on .
Theorem 31. The RKHS of the kernel consists of all functions on for which there are sequences of functions in and of functions in so that
and the norm is given by
where the minimum is taken over the set of all sequences and satisfying (44).
3.3 Examples of RKHS. Bergman and Hardy spaces
Bergman space. The space of all analytic functions on for which
is satisfied is called the Bergman space on and denoted by .
Remark. is a RKHS with respect to the inner product
and its kernel is called the Bergman kernel on and denoted by
Bergman kernel for the unit disc. The Bergman kernel for the open unit disc is given by
Bergman kernel of a simply connected domain. The Bergman kernel of a simply connected domain is given by
where is any conformal mapping function from onto
Theorem 32. A conformal mapping from to can be recovered from the Bergman kernel of
Jordan curve. A Jordan curve is a continuous image of in .
Green function. A Green function of is a function harmonic in except at where it has logarithmic singularity, and continuous in the closure with boundary values for all where is a finitely connected domain of the complex plane.
Theorem 33. Let be a finitely connected domain bounded by analytic Jordan curves, and let be the Green’s function of Then the Bergman kernel function is
Hardy space. The closed linear span of in is called the (Hilbert type) Hardy space on and is denoted by Here .
Remark. belongs to the Hardy space if and only if it is orthonormal to all (), that is, all Fourier coefficients of with negative indices vanish. Then we have
Szegö kernel. The kernel , or its analytic extension is 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 . The Hilbert space and functional analysis parts of this chapter are based on the books by J.B. Conway  and R.G. Douglas . On the other hand, the reproducing kernel part is based on the lecture notes of T. Ando  and N. Aronszajn , the book of S. Saitoh and Y. Sawano , and the book of B. Okutmustur and A. Gheondea . Moreover, the details of Bergman and Hardy spaces are widely explained in the books [7, 8, 9].