A Survey on Hilbert Spaces and Reproducing Kernels

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.


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.

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 , then it follows that i. if x, y, z ∈ H, then ii. if k is scalar, then kx ∈ H: Inner product. Let H be a linear space over the complex field . An inner product on H is a two variable function Pre-Hilbert space. A pre-Hilbert space H 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 H denoted by ∥ Á ∥ is defined by where f ∈ H and Á, Á h i ¼ Á, Á h i 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 , g ∈ H, and λ ∈ , we have • ∥ f ∥ ≥ 0. (Observe that the equality occurs only if f ¼ 0).
• ∥λf ∥ ¼ |λ|∥ f ∥: Schwarz inequality. For all f , g ∈ H, it follows that In case if f and g are linearly dependent, then the inequality becomes equality. Triangle inequality. For all f , g ∈ H, it follows that In case if f and g are linearly dependent, then the inequality becomes equality. Polarization identity. For all f , g ∈ H, it follows that Parallelogram identity. For all f , g ∈ H, it follows that • d x, y ð Þ≥ 0 and d x, y ð Þ ¼ 0 only if x ¼ y; for all x, y, z ∈ X. Moreover the space X, d ð Þis 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, h ∈ H and for all λ ∈ , where d satisfies all requirements to be a metric, we have and equality occurs only if f ¼ g: 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 f ð Þ n ≥ 0 in the pre-Hilbert space H converges strongly to f if the condition is satisfied.

Introduction to linear operators
Linear operator. A map L from a linear space to another linear space is called linear operator if is satisfied for all α, β ∈  and for all f , g ∈ H. 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 Theorem 1. Let L be a linear operator from the pre-Hilbert spaces H to G: Then the followings are mutually equivalent: i. L is continuous.
ii. L is bounded, that is, iii. L is Lipschitz continuous, that is, where 0 ≤ λ < ∞ and f , g ∈ H: Some properties of linear operators. Let B H, G ð Þbe the collection of all continuous linear operators from the pre-Hilbert spaces H to G. Then Þis a linear space with respect to the natural addition and scalar multiplication satisfying where L and M are linear operators, f ∈ H and α, β ∈ : In addition, are also satisfied.

Hilbert spaces and linear operators
Linear form (or linear functional). A linear operator from the pre-Hilbert space H to the scalar field  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 f n is a Cauchy sequence in H, that is, if 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 spaceH so that 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 f ∈ H, g ∈ M and 0 ≤ λ < ∞, if a linear operator L from M to G satisfies 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 Ω, μ ð Þdenotes a measure space so that Ω is the union of subsets of finite positive measure and Then L 2 Ω, μ ð Þis a Hilbert space with respect to the inner product

Theorem 4 (F. Riesz). For each continuous linear functional φ on a Hilbert
Theorem 5. Let M be a closed subspace of a Hilbert space H: Then the algebraic direct sum relation is satisfied. In other words, ∀f ∈ H can be uniquely written by In Remark. In a Hilbert space, the closed linear span of any subset A of a Hilbert 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, Note. If I is the identity operator on H, then I À P M denotes the orthogonal projection to M ⊥ , and the relation is satisfied for all f ∈ H. Weak topology. The weakest topology that makes continuous all linear functionals of the form f ↦ f , g h iis called the weak topology of a Hilbert space H. Note. If f ∈ H, then with respect to the weak topology, a fundamental system of neighborhoods of f is composed of subsets of the form where A is a finite subset of H and ϵ > 0. Then a directed net f λ È É converges weakly to f 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 B H, G ð Þof continuous linear operators from H to G. In addition, a directed net L λ f g converges weakly to L if Operator strong topology.
then it is strongly bounded, that is, sup f ∈ A ∥ f ∥ < ∞: 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 L ⊆ B H, G ð Þare mutually equivalent: (i) L is weakly bounded; that is, for f ∈ H, g ∈ G, we have (iii) L is norm bounded (or uniformly bounded); that is, Theorem 10. A linear operator L from the Hilbert spaces H to G is said to be closed if its graph is a closed subspace of the direct sum space H ⊕ G, that is, whenever n ! ∞, |f n À f ∥ ! 0 in H and ∥Lf n À g∥ ! 0 in G ) g ¼ 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 Â G !  is a sesqui-linear form (or sesquilinear function) if for f , h ∈ H, g, k ∈ G and α, β ∈ , are satisfied where H and G are Hilbert spaces.
Remark. If L ∈ B H, G ð Þ, then the sesqui-linear form Φ defined by is bounded in the sense that where λ ≥ ∥L∥: Hence f ↦f 0 becomes linear, and as a result we obtain is called the adjoint of L. Remark. By the definitions of L and L * , it follows that Isometric property. The adjoint operation is isometric if Remark. Let H, G, and K be Hilbert spaces and K ∈ B G, K ð Þand L ∈ B H, G ð Þbe given. Then where Ker L ð Þ is the kernel of L and Ran L ð Þ is the range of L. Theorem 12. If L, M ∈ B H, G ð Þ, then the following statements are mutually equivalent.
ii. There exists K ∈ B H ð Þ such that M ¼ LK: iii. There exists 0 ≤ λ < ∞ such that ∥M * g∥ ≤ λ∥L * g∥ for g ∈ G: Quadric form. Let H be a Hilbert space. A function and are satisfied. 8

Functional Calculus
Note. If L ∈ B H ð Þ, the quadratic form φ on H is defined by and it is bounded where λ ≥ ∥L∥: Remark. The sesqui-linear form Φ associated with L can be recovered from the quadratic form φ by the equation for all f , g ∈ H: Self-adjoint operator. A continuous linear operator L on a Hilbert space H is 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 i≥ 0 for all f ∈ H: If Lf , f h i¼ 0 only when f ¼ 0, then L is said to be strictly positive (or, strictly positive definite).
Note. For any positive operator L ∈ B H ð Þ, the Schwarz inequality holds in the following sense 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 if and only if the continuous linear operator 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 L 1=2 , such that L 1=2 À Á 2 ¼ L: Modulus operator. The square root of the positive definite operator L * L 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 is satisfied, that is, it preserves the norm. Note. Eq. (32) implies that a continuous linear operator U is isometric if and only if U * U ¼ I H ; in other words, The spaces KerU ð Þ ⊥ and Ran U ð Þ 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 whereL is a positive definite operator and U is a partial isometry with initial space the closure of RanL À Á .

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

Definition and fundamental properties
Reproducing kernels. Let H be a Hilbert space of functions on a nonempty set X with the inner product f , are satisfied.
Note. Let K be a reproducing kernel. Applying (35) to the function K x at y, we get Then, for any x ∈ X, we obtain Note. Observe that the subset K x f g x ∈ X is total in H, that is, its closed linear span coincides with H. This follows from the fact that, if f ∈ H and f ⊥K x for all x ∈ X, then and hence f is the 0 element in H. As a result, 0 f g ⊥ ¼ H. Hermitian and positive definite kernel. Let X be an arbitrary set and K be a kernel on X, that is, K : X Â X ! . The kernel K is called Hermitian if for any finite set of points y 1 , … , y n È É ⊆ X, we have

RKHS.
It is called positive definite, if for any complex numbers ϵ 1 , … , ϵ n , we have ϵ j ϵ i K y j , y i ≥ 0: Note. From the previous inequality, it follows that for any finitely supported family of complex numbers ϵ x f g x ∈ X , we have iv. Let x 0 ∈ X. Then the following statements are equivalent: a. K x 0 , x 0 ð Þ¼0.
b. K y, x 0 ð Þ¼0 for all y ∈ X.
c. f x 0 ð Þ ¼ 0 for all f ∈ H: Theorem 21. For any positive definite kernel K on X, there exists a unique Hilbert space H K of functions on X with reproducing kernel K.
Theorem 22. Every sequence of functions f n À Á n ≥ 1 that converges strongly to a function f in H K X ð Þ converges also in the pointwise sense, i.e., for any point x ∈ X, In addition, this convergence is uniform on every subset of X on which x↦K x, x ð Þ is bounded. Theorem 23. A complex valued function g on X belongs to the reproducing kernel Hilbert space H K X ð Þ if and only if there exists 0 ≤ λ < ∞ such that, ∥g∥ coincides with the minimum of all such λ. Theorem 24. If K 1 ð Þ y, x ð Þand K 2 ð Þ y, x ð Þare two positive definite kernels on X, then the following statements are mutually equivalent: ii. There exists 0 ≤ λ < ∞ such that Note. For any map φ from a set X to a Hilbert space H, with the notation x↦φ x , a kernel K can be defined by Theorem 25. Let φ : X↦H be an arbitrary map and for x, y ∈ X let K be defined as 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 Then Ran T ð Þ coincides with H K X ð Þ and