A Survey on Hilbert Spaces and Reproducing Kernels

*Baver Okutmuştur*

### **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.

#### **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 H over the field , then it follows that

i. if *x*, *y*, *z*∈ H, then

$$
\infty + \mathcal{y} = \mathcal{y} + \mathfrak{x} \in \mathcal{H}, \quad \mathfrak{x} + (\mathfrak{y} + \mathfrak{z}) = (\mathfrak{x} + \mathfrak{y}) + \mathfrak{z} \in \mathcal{H};
$$

• *d x*ð Þ , *y* ≥0 and *d x*ð Þ¼ , *y* 0 only if *x* ¼ *y*;

*A Survey on Hilbert Spaces and Reproducing Kernels DOI: http://dx.doi.org/10.5772/intechopen.91479*

• *d f* ð Þ , *g* ≥0 and equality occurs only if *f* ¼ *g:*

for all *x*, *y*, *z*∈ *X*. Moreover the space ð Þ *X*, *d* 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*, *h*∈ H and for all *λ*∈ , where *d* satisfies all requirements to

**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 ð Þ*f <sup>n</sup>*≥<sup>0</sup> in the pre-Hilbert space H converges strongly to *f* if the condition

∥*f <sup>n</sup>* � *f* ∥ ! 0 whenever *n* ! ∞

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

*L*ð Þ¼ *αf* þ *βg αLf* þ *βLg*

**Continuous operator.** An operator *L* is said to be continuous if it is continuous

**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

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

supf g ∥*Lf* ∥*<sup>G</sup>* : ∥ *f* ∥*<sup>H</sup>* ≤ *k* < ∞

<sup>∥</sup>*L*<sup>∥</sup> <sup>¼</sup> sup <sup>∥</sup>*Lf* <sup>∥</sup>G*=*<sup>∥</sup> *<sup>f</sup>* <sup>∥</sup><sup>H</sup> : <sup>0</sup> 6¼ *<sup>f</sup>* <sup>∈</sup> <sup>H</sup> *:* (5)

at each point of its domain. Notice that the domain and range spaces must be

• *d x*ð Þ¼ , *y d y* ð Þ , *x* ;

be a metric, we have

• *d f* ð Þ¼ , *g d g*ð Þ , *f :*

• *d f* ð Þ , *g* ≤*d f* ð Þþ , *h d h*ð Þ , *g :*

• *d f* ð Þ¼ � *h*, *g* � *h d f* ð Þ , *g* .

**2.2 Introduction to linear operators**

convenient for appropriate topologies.

the followings are mutually equivalent:

i. *L* is continuous.

for 0≤ *k*< ∞*:*

**63**

ii. *L* is bounded, that is,

is satisfied for all *α*, *β* ∈ and for all *f*, *g* ∈ H.

• *d*ð Þ¼ *λf*, *λg* ∣*λ*∣ � *d f* ð Þ , *g :*

is satisfied.

*linear operator* if

∥*L*∥ and it is defined by

• *d x*ð Þ , *y* ≤*d x*ð Þþ , *z d z*ð Þ , *y* ;

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

$$
\langle \cdot, \cdot \rangle : \mathcal{H} \times \mathcal{H} \to \mathbb{C}, \text{ satisfying } \cdot
$$

$$\text{i. } \langle f, \mathbf{g} \rangle = \overline{\langle \mathbf{g}, f \rangle} \text{ for } f, \mathbf{g} \in \mathcal{H}.$$

ii. h i *αf* þ *βg*, *h* ¼ *α*h i *f*, *h* þ *β*h i *g*, *h* and h i *f*, *αg* þ *βh* ¼ *α*h i *f*, *g* þ *β*h i *f*, *h* for *α*, *β* ∈ and *f*, *g*, *h*∈ H*:*

$$\text{iii.}\ \langle f, f \rangle \ge 0 \text{ for} \\ f \in \mathcal{H} \text{ and } \ \langle f, f \rangle = 0 \\ \Leftrightarrow f = \mathbf{0}.$$

**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 ∥ � ∥ is defined by

$$\|f\| = \langle f, f \rangle^{1/2} \text{ or } \|f\|\_{\mathcal{H}} = \langle f, f \rangle\_{\mathcal{H}}^{1/2}$$

where *<sup>f</sup>* <sup>∈</sup> <sup>H</sup> and h i �, � ¼ �h i , � <sup>H</sup> denote the inner product on <sup>H</sup>. 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


**Schwarz inequality.** For all *f*, *g* ∈ H, it follows that

$$|\langle f, \mathbf{g} \rangle| \le \|f\| \|\mathbf{g}\|. \tag{1}$$

In case if *f* and *g* are linearly dependent, then the inequality becomes equality. **Triangle inequality.** For all *f*, *g* ∈ H, it follows that

$$\|f+\mathbf{g}\| \le \|f\| + \|\mathbf{g}\|. \tag{2}$$

In case if *f* and *g* are linearly dependent, then the inequality becomes equality. **Polarization identity.** For all *f*, *g* ∈ H, it follows that

$$\langle f, \mathbf{g} \rangle = \frac{1}{4} \left( \|f + \mathbf{g}\|^2 - \|f - \mathbf{g}\|^2 + i\|f + i\mathbf{g}\|^2 - \|f - i\mathbf{g}\|^2 \right) \text{ for} f, \mathbf{g} \in \mathcal{H}. \tag{3}$$

**Parallelogram identity.** For all *f*, *g* ∈ H, it follows that

$$\|\|f+\mathbf{g}\|\|^2 + \|f-\mathbf{g}\|\|^2 = 2\|f\|\|^2 + 2\|\mathbf{g}\|\|^2. \tag{4}$$

**Metric.** A metric on a set *X* is a function *d*: *X* � *X* ! satisfying the properties.

*A Survey on Hilbert Spaces and Reproducing Kernels DOI: http://dx.doi.org/10.5772/intechopen.91479*


i. if *x*, *y*, *z*∈ H, then

*Functional Calculus*

ii. if *k* is scalar, then *kx*∈ H*:*

*product* on H is a two variable function

i. h i *f*, *g* ¼ h i *g*, *f* for *f*, *g* ∈ H*:*

with an inner product defined on it.

denoted by ∥ � ∥ is defined by

• ∥*λf* ∥ ¼ ∣*λ*∣∥ *f* ∥*:*

h i *<sup>f</sup>*, *<sup>g</sup>* <sup>¼</sup> <sup>1</sup>

**62**

*β*h i *f*, *h* for *α*, *β* ∈ and *f*, *g*, *h*∈ H*:*

iii. h i *f*, *f* ≥0 for *f* ∈ H and h i *f*, *f* ¼ 0 ⇔ *f* ¼ 0*:*

<sup>∥</sup> *<sup>f</sup>* <sup>∥</sup> <sup>¼</sup> h i *<sup>f</sup>*, *<sup>f</sup>* <sup>1</sup>*=*<sup>2</sup>

**Properties of norm.** For all *f*, *g* ∈ H, and *λ*∈ , we have

• ∥ *f* ∥ ≥0. (Observe that the equality occurs only if *f* ¼ 0).

**Schwarz inequality.** For all *f*, *g* ∈ H, it follows that

**Triangle inequality.** For all *f*, *g* ∈ H, it follows that

**Polarization identity.** For all *f*, *g* ∈ H, it follows that

**Parallelogram identity.** For all *f*, *g* ∈ H, it follows that

where *<sup>f</sup>* <sup>∈</sup> <sup>H</sup> and h i �, � ¼ �h i , � <sup>H</sup> denote the inner product on <sup>H</sup>. The corresponding space is called as the inner product space or the norm space.

*x* þ *y* ¼ *y* þ *x*∈ H, *x* þ ð Þ¼ *y* þ *z* ð Þþ *x* þ *y z*∈ H;

**Inner product.** Let H be a linear space over the complex field . An *inner*

h i �, � : H�H! , satisfying

**Pre-Hilbert space.** A *pre-Hilbert space* H is a linear space over the complex field

or <sup>∥</sup> *<sup>f</sup>* <sup>∥</sup><sup>H</sup> <sup>¼</sup> h i *<sup>f</sup>*, *<sup>f</sup>* <sup>1</sup>*=*<sup>2</sup>

H

∣h i *f*, *g* ∣≤∥ *f* ∥∥*g*∥*:* (1)

∥ *f* þ *g*∥≤∥ *f* ∥ þ ∥*g*∥*:* (2)

*:* (4)

**Norm space or inner product space.** A norm on an inner product space H

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

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

<sup>∥</sup> *<sup>f</sup>* <sup>þ</sup> *<sup>g</sup>*∥<sup>2</sup> <sup>þ</sup> <sup>∥</sup> *<sup>f</sup>* � *<sup>g</sup>*∥<sup>2</sup> <sup>¼</sup> <sup>2</sup><sup>∥</sup> *<sup>f</sup>* <sup>∥</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>∥*g*∥<sup>2</sup>

**Metric.** A metric on a set *X* is a function *d*: *X* � *X* ! satisfying the properties.

<sup>4</sup> <sup>∥</sup> *<sup>f</sup>* <sup>þ</sup> *<sup>g</sup>*∥<sup>2</sup> � <sup>∥</sup> *<sup>f</sup>* � *<sup>g</sup>*∥<sup>2</sup> <sup>þ</sup> *<sup>i</sup>*<sup>∥</sup> *<sup>f</sup>* <sup>þ</sup> *ig*∥<sup>2</sup> � <sup>∥</sup> *<sup>f</sup>* � *ig*∥<sup>2</sup> for *<sup>f</sup>*, *<sup>g</sup>* <sup>∈</sup> <sup>H</sup>*:* (3)

ii. h i *αf* þ *βg*, *h* ¼ *α*h i *f*, *h* þ *β*h i *g*, *h* and h i *f*, *αg* þ *βh* ¼ *α*h i *f*, *g* þ

• *d x*ð Þ , *y* ≤*d x*ð Þþ , *z d z*ð Þ , *y* ;

for all *x*, *y*, *z*∈ *X*. Moreover the space ð Þ *X*, *d* 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*, *h*∈ H and for all *λ*∈ , where *d* satisfies all requirements to be a metric, we have

• *d f* ð Þ , *g* ≥0 and equality occurs only if *f* ¼ *g:*

$$\bullet \ d(f, \mathbf{g}) = d(\mathbf{g}, f) \,.$$

• *d f* ð Þ , *g* ≤*d f* ð Þþ , *h d h*ð Þ , *g :*

$$\bullet \ d(f - h, \mathbf{g} - h) = d(f, \mathbf{g}) \cdot \mathbf{g}$$

• *d*ð Þ¼ *λf*, *λg* ∣*λ*∣ � *d f* ð Þ , *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 ð Þ*f <sup>n</sup>*≥<sup>0</sup> in the pre-Hilbert space H converges strongly to *f* if the condition

$$\|f\_n - f\| \to 0 \text{ whenever } n \to \infty$$

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

$$L(qf + \beta \mathbf{g}) = aLf + \beta \mathbf{Lg}$$

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

$$\|L\| = \sup\{\|Lf\|\_{\mathcal{G}}/\|f\|\_{\mathcal{H}} : \mathbf{0} \neq f \in \mathcal{H}\}.\tag{5}$$

**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,

$$\sup \{ \|Lf\|\_{G} : \|f\|\_{H} \le k \} \ll \infty$$

for 0≤ *k*< ∞*:*

iii. *L* is Lipschitz continuous, that is,

$$\|Lf - L\mathbf{g}\|\_{G} \le \lambda \|f - \mathbf{g}\|\_{\mathcal{H}},$$

**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

**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

then *L* is uniquely extended to a continuous linear operator from M to G with

**Theorem 3.** Let ð Þ Ω, *μ* denotes a measure space so that Ω is the union of subsets

ð Þ Ω, *μ* is 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

H¼M⊕M<sup>⊥</sup>

**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,

<sup>A</sup><sup>⊥</sup> <sup>¼</sup> f g<sup>0</sup> *:*

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

**Orthogonal projection.** If <sup>M</sup> is a closed subspace of <sup>H</sup>, the map *<sup>f</sup>*↦*f*<sup>M</sup> gives a linear operator from H to M with norm ≤ 1. We call this operator as the *orthogonal*

ð Ω

norm ≤*λ* where the norm coincides with the minimum of such *λ*.

ð Ω

h i *f*, *g* ≔

is satisfied. In other words, ∀*f* ∈ H can be uniquely written by

In addition, <sup>∥</sup>*f*M<sup>∥</sup> coincides with the distance from *<sup>f</sup>* to <sup>M</sup><sup>⊥</sup>

*:*

space H, there exists uniquely *g* ∈ H such that

<sup>∣</sup>h i *Lf*, *<sup>g</sup>* <sup>G</sup>∣≤∣*λ*∣∥ *<sup>f</sup>* <sup>∥</sup>H∥*g*∥G, (6)

ð Þ Ω, *μ* consists of all measurable functions *f*ð Þ *ω* on

j j *<sup>f</sup>*ð Þ *<sup>ω</sup>* <sup>2</sup> *<sup>d</sup>μ ω*ð Þ<sup>&</sup>lt; <sup>∞</sup>*:* (7)

*f*ð Þ *ω g*ð Þ *ω dμ ω*ð Þ*:* (8)

*φ*ð Þ¼ *f* h i *f*, *g for f* ∈ H*:* (9)

*<sup>f</sup>* <sup>¼</sup> *<sup>f</sup>*<sup>M</sup> <sup>þ</sup> *<sup>f</sup>*M<sup>⊥</sup> with *<sup>f</sup>*<sup>M</sup> <sup>∈</sup>M, *<sup>f</sup>*M<sup>⊥</sup> <sup>∈</sup>M<sup>⊥</sup>*:* (10)

<sup>∥</sup>*f*M<sup>∥</sup> <sup>¼</sup> min <sup>∥</sup> *<sup>f</sup>* � *<sup>g</sup>*<sup>∥</sup> : *<sup>g</sup>* <sup>∈</sup>M<sup>⊥</sup> � �*:* (11)

operator from H to G with preserving norm.

*A Survey on Hilbert Spaces and Reproducing Kernels DOI: http://dx.doi.org/10.5772/intechopen.91479*

of finite positive measure and *L*<sup>2</sup>

satisfies

Ω such that

Then *L*<sup>2</sup>

direct sum relation

space <sup>H</sup> coincides with *<sup>A</sup>*<sup>⊥</sup> � �<sup>⊥</sup>

linear combinations of elements of A.

*projection* to M and denote it by *P*M.

**65**

where 0 ≤*λ*< ∞ and *f*, *g* ∈ H*:*

**Some properties of linear operators.** Let *B*ð Þ H, G be the collection of all continuous linear operators from the pre-Hilbert spaces H to G. Then

• *B*ð Þ H, G is a linear space with respect to the natural addition and scalar multiplication satisfying

$$(aL + \beta \mathbf{M})f = aLf + \beta \mathbf{M}f,$$

where *L* and *M* are linear operators, *f* ∈ H and *α*, *β* ∈ *:*


$$(KL)f = K(Lf)\text{ for }f \in \mathcal{H} \in B(\mathcal{H}, \mathbb{K}).$$

In addition,


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 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 <sup>n</sup>* is a Cauchy sequence in H, that is, if

∥*f <sup>n</sup>* � *f <sup>m</sup>*∥ ! 0 whenever *n*, *m* ! ∞,

then there is *f* ∈ H such that

$$\|f\_n - f\| \to \mathbf{0} \text{ whenever } n \to \infty.$$

**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 <sup>H</sup><sup>~</sup> so that

$$\langle f, \mathfrak{g} \rangle\_{\hat{\mathcal{H}}} = \langle f, \mathfrak{g} \rangle\_{\mathcal{H}} \text{ for} \\ f, \mathfrak{g} \in \mathcal{H}$$

is called *completion*.

*A Survey on Hilbert Spaces and Reproducing Kernels DOI: http://dx.doi.org/10.5772/intechopen.91479*

iii. *L* is Lipschitz continuous, that is,

where 0 ≤*λ*< ∞ and *f*, *g* ∈ H*:*

*Functional Calculus*

multiplication satisfying

i. *K*ð Þ¼ *ξL* þ *ζM ξKL* þ *ζKM*

iii. ∥*L* þ *M*∥≤∥*L*∥ þ ∥*M*∥ and

**2.3 Hilbert spaces and linear operators**

then there is *f* ∈ H such that

is called *completion*.

**64**

ii. ∥*ξL*∥ ¼ ∣*ξ*∣ � ∥*L*∥

iv. ∥*KL*∥≤∥*K*∥∥*L*∥.

are also satisfied.

In addition,

∥*Lf* � *Lg*∥*<sup>G</sup>* ≤ *λ*∥ *f* � *g*∥H,

**Some properties of linear operators.** Let *B*ð Þ H, G be the collection of all con-

ð Þ *αL* þ *βM f* ¼ *αLf* þ *βMf*,

• If K is another pre-Hilbert space, *L*∈ *B*ð Þ H, G and *K* ∈*B*ð Þ G, K . Then the product

ð Þ *KL f* ¼ *K Lf* ð Þ for *f* ∈ H ∈*B*ð Þ H, K *:*

**Linear form (or linear functional).** A linear operator from the pre-Hilbert

∥*f <sup>n</sup>* � *f <sup>m</sup>*∥ ! 0 whenever *n*, *m* ! ∞,

∥*f <sup>n</sup>* � *f* ∥ ! 0 whenever *n* ! ∞*:*

**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

h i *<sup>f</sup>*, *<sup>g</sup>* <sup>H</sup><sup>~</sup> <sup>¼</sup> h i *<sup>f</sup>*, *<sup>g</sup>* <sup>H</sup> for *<sup>f</sup>*, *<sup>g</sup>* <sup>∈</sup> <sup>H</sup>

**Hilbert spaces.** A pre-Hilbert space H is said to be a *Hilbert space* if it is complete

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

in metric. In other words if *f <sup>n</sup>* is a Cauchy sequence in H, that is, if

embedded as a dense subspace of a Hilbert space <sup>H</sup><sup>~</sup> so that

• *B*ð Þ H, G is a linear space with respect to the natural addition and scalar

tinuous linear operators from the pre-Hilbert spaces H to G. Then

where *L* and *M* are linear operators, *f* ∈ H and *α*, *β* ∈ *:*

• Whenever H¼G, then *B*ð Þ H, G is denoted by *B*ð Þ H .

**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

$$|\langle Lf, \mathbf{g} \rangle\_{\mathcal{G}}| \le |\lambda| \|f\|\_{\mathcal{H}} \|\mathbf{g}\|\_{\mathcal{G}},\tag{6}$$

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 ð Þ Ω, *μ* denotes a measure space so that Ω is the union of subsets of finite positive measure and *L*<sup>2</sup> ð Þ Ω, *μ* consists of all measurable functions *f*ð Þ *ω* on Ω such that

$$\int\_{\Omega} |f(\alpha)|^2 d\mu(\alpha) < \infty. \tag{7}$$

Then *L*<sup>2</sup> ð Þ Ω, *μ* is a Hilbert space with respect to the inner product

$$\mathcal{L}\langle f, \mathbf{g} \rangle \coloneqq \int\_{\Omega} f(\boldsymbol{\alpha}) \overline{\mathbf{g}(\boldsymbol{\alpha})} d\boldsymbol{\mu}(\boldsymbol{\alpha}).\tag{8}$$

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

$$
\rho(f) = \langle f, \mathfrak{g} \rangle \, for \, f \in \mathcal{H}. \tag{9}
$$

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

$$\mathcal{H} = \mathcal{M} \oplus \mathcal{M}^\perp$$

is satisfied. In other words, ∀*f* ∈ H can be uniquely written by

$$f = f\_{\mathcal{M}} + f\_{\mathcal{M}^\perp} \text{ with } f\_{\mathcal{M}} \in \mathcal{M}, f\_{\mathcal{M}^\perp} \in \mathcal{M}^\perp. \tag{10}$$

In addition, <sup>∥</sup>*f*M<sup>∥</sup> coincides with the distance from *<sup>f</sup>* to <sup>M</sup><sup>⊥</sup>

$$\|\!\|f\!\!\!\_{\mathcal{M}}\| = \min\left\{ \|\!\|f-\mathbf{g}\|:\mathbf{g}\in\mathcal{M}^{\perp} \right\}.\tag{11}$$

**Remark.** In a Hilbert space, the closed linear span of any subset *A* of a Hilbert space <sup>H</sup> coincides with *<sup>A</sup>*<sup>⊥</sup> � �<sup>⊥</sup> *:*

**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,

$$\mathcal{A}^\perp = \{\mathbf{0}\}.$$

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 <sup>M</sup> is a closed subspace of <sup>H</sup>, the map *<sup>f</sup>*↦*f*<sup>M</sup> 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 *P*M.

**Note.** If *I* is the identity operator on H, then *I* � *P*<sup>M</sup> denotes the orthogonal projection to <sup>M</sup><sup>⊥</sup>, and the relation

$$\|f\|^2 = \|P\_{\mathcal{M}}f\|^2 + \|(I - P\_{\mathcal{M}})f\|^2 \tag{12}$$

**Theorem 8.** If H and G are Hilbert spaces and *L* is a linear operator from H to G,

∣h i *Lf*, *g* ∣< ∞

∥*Lf* ∥< ∞*:*

∥*L*∥< ∞*:*

*GL* ≔ f g *f* ⊕ *Lf* : *f* ∈ H (14)

ð Þi Φð Þ¼ *αf* þ *βh*, *g α*Φð Þþ *f*, *g β*Φð Þ *h*, *g* (15) ð Þii Φð Þ¼ *f*, *αg* þ *βk α*Φð Þþ *f*, *g β*Φð Þ *f*, *k* (16)

∣Φð Þ *f*, *g* ∣ ≤ *λ*∥ *f* ∥H∥*g*∥<sup>G</sup> for *f* ∈ H, *g* ∈ G, (18)

Φð Þ¼ *f*, *g* h i *Lf*, *g <sup>G</sup>* (17)

**Theorem 10.** A linear operator *L* from the Hilbert spaces H to G is said to be

is a closed subspace of the direct sum space H ⊕ G, that is, whenever *n* ! ∞,

∣*f <sup>n</sup>* � *f* ∥ ! 0 *in* H *and* ∥*Lf <sup>n</sup>* � *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

**Sesqui-linear form.** A function Φ : H�G! is a *sesqui-linear form* (or *sesqui-*

**Remark.** If a sesqui-linear form Φ satisfies the condition (18), then for *f* ∈ H,

*g*↦Φð Þ *f*, *g*

is continuous on G*:* If we apply the Riesz theorem, then there exists uniquely

**Theorem 9.** Let H and G be Hilbert spaces. Then the following statements for

then the strong continuity and weak continuity for *L* are equivalent.

(i) **L** is weakly bounded; that is, for *f* ∈ H, *g* ∈G, we have

(ii) **L** is strongly bounded; that is, for *f* ∈ H, we have

(iii) **L** is norm bounded (or uniformly bounded); that is,

another Hilbert space G, then it is continuous.

*linear function*) if for *f*, *h*∈ H, *g*, *k*∈G and *α*, *β* ∈ ,

are satisfied where H and G are Hilbert spaces.

is bounded in the sense that

where *λ*≥ ∥*L*∥*:*

the linear functional

∈G satisfying

*f* 0

**67**

**Remark.** If *L*∈*B*ð Þ H, G , then the sesqui-linear form Φ defined by

sup *L*∈ **L**

> sup *L*∈ **L**

> > sup *L*∈ **L**

**L**⊆ *B*ð Þ H, G are mutually equivalent:

*A Survey on Hilbert Spaces and Reproducing Kernels DOI: http://dx.doi.org/10.5772/intechopen.91479*

*closed* if its *graph*

is satisfied for all *f* ∈ H.

**Weak topology.** The weakest topology that makes continuous all linear functionals of the form *f*↦h i *f*, *g* is 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

$$U(f; A, \epsilon) = \{ h : |\langle f, g \rangle - \langle h, g \rangle| < \epsilon \text{ for } g \in \mathcal{A} \},$$

where A is a finite subset of H and ϵ>0. Then a directed net *f <sup>λ</sup>* converges weakly to *f* if and only if

$$
\langle f\_{\lambda}, \mathbf{g} \rangle \stackrel{\lambda}{\to} \langle f, \mathbf{g} \rangle \text{ for all } \mathbf{g} \in \mathcal{H}.
$$

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

$$L \mapsto \langle Lf, g \rangle \text{ for} \\ f \in \mathcal{H}, g \in \mathcal{G}$$

is called *the operator weak topology* in the space *B*ð Þ H, G of continuous linear operators from H to G. In addition, a directed net f g *L<sup>λ</sup>* converges weakly to *L* if

$$
\langle Lf, \mathfrak{g} \rangle \xrightarrow{\mathring{\lambda}} \langle Lf, \mathfrak{g} \rangle .
$$

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

$$L \mapsto Lf \text{ for } f \in \mathcal{H}$$

is called *the operator strong topology*. Moreover a directed net f g *L<sup>λ</sup>* converges strongly to *L* if

$$\|L\_{\vec{\mathbb{A}}}f - Lf\| \stackrel{\vec{\mathbb{A}}}{\to} \mathbf{0} \text{ for all } f \in \mathcal{H}.$$

**Theorem 6**. Let H and G be Hilbert spaces and *B*ð Þ H, G be a continuous linear operator from H to G. Then


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

$$\sup\_{f \in A} |\langle f, \mathbf{g} \rangle| < \infty \text{ for } \mathbf{g} \in \mathcal{H}, \tag{13}$$

then it is strongly bounded, that is, sup *<sup>f</sup>* <sup>∈</sup> *<sup>A</sup>*∥ *f* ∥< ∞*:*

*A Survey on Hilbert Spaces and Reproducing Kernels DOI: http://dx.doi.org/10.5772/intechopen.91479*

**Note.** If *I* is the identity operator on H, then *I* � *P*<sup>M</sup> denotes the orthogonal

**Weak topology.** The weakest topology that makes continuous all linear func-

**Note.** If *f* ∈ H, then with respect to the weak topology, a fundamental system of

h i *f*, *g* for all *g* ∈ H*:*

h i *Lf*, *g :*

*<sup>λ</sup>* 0 for all *<sup>f</sup>* <sup>∈</sup> <sup>H</sup>*:*

*U f* ð Þ¼ ; *A*, ϵ f g *h* :jh i *f*, *g* � h ij *h*, *g* <ϵ for *g* ∈ A ,

**Operator weak topology.** The weakest topology that makes continuous all

*L*↦h i *Lf*, *g* for *f* ∈ H, *g* ∈ G

is called *the operator weak topology* in the space *B*ð Þ H, G of continuous linear operators from H to G. In addition, a directed net f g *L<sup>λ</sup>* converges weakly to *L* if

**Operator strong topology.** The weakest topology that makes continuous all

*L*↦*Lf* for *f* ∈ H

is called *the operator strong topology*. Moreover a directed net f g *L<sup>λ</sup>* converges

**Theorem 6**. Let H and G be Hilbert spaces and *B*ð Þ H, G be a continuous linear

**Theorem 7.** Let H be a Hilbert space and *A* ⊆ H*:* Then if *A* is weakly bounded in

∣h i *f*, *g* ∣< ∞ for *g* ∈ H, (13)

∥*Lλf* � *Lf* ∥ !

• the closed unit ball *U* ≔ f g *f* : ∥ *f* ∥ ≤ 1 of H is weakly compact;

• the closed unit ball f g *L* : ∥*L*∥ ≤1 of *B*ð Þ H, G is weakly compact.

sup *f* ∈ *A*

then it is strongly bounded, that is, sup *<sup>f</sup>* <sup>∈</sup> *<sup>A</sup>*∥ *f* ∥< ∞*:*

h i *Lλf*, *g* ! *λ*

tionals of the form *f*↦h i *f*, *g* is called *the weak topology* of a Hilbert space H.

where A is a finite subset of H and ϵ>0. Then a directed net *f <sup>λ</sup>*

*<sup>f</sup> <sup>λ</sup>*, *<sup>g</sup>* ! *λ*

neighborhoods of *f* is composed of subsets of the form

<sup>∥</sup> *<sup>f</sup>* <sup>∥</sup><sup>2</sup> <sup>¼</sup> <sup>∥</sup>*P*M*<sup>f</sup>* <sup>∥</sup><sup>2</sup> <sup>þ</sup> <sup>∥</sup>ð Þ *<sup>I</sup>* � *<sup>P</sup>*<sup>M</sup> *<sup>f</sup>* <sup>∥</sup><sup>2</sup> (12)

converges

projection to <sup>M</sup><sup>⊥</sup>, and the relation

*Functional Calculus*

is satisfied for all *f* ∈ H.

weakly to *f* if and only if

linear functionals of the form

linear operators of the form

operator from H to G. Then

strongly to *L* if

the sense

**66**

**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*ð Þ H, G are mutually equivalent:

(i) **L** is weakly bounded; that is, for *f* ∈ H, *g* ∈G, we have

$$\sup\_{L \in \mathcal{L}} |\langle Lf, \emptyset \rangle| < \infty$$

(ii) **L** is strongly bounded; that is, for *f* ∈ H, we have

$$\sup\_{L \in \mathbf{L}} \|Lf\| < \infty$$

(iii) **L** is norm bounded (or uniformly bounded); that is,

$$\sup\_{L \in \mathcal{L}} \|L\| < \infty.$$

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

$$G\_L \coloneqq \{ f \oplus Lf : f \in \mathcal{H} \}\tag{14}$$

is a closed subspace of the direct sum space H ⊕ G, that is, whenever *n* ! ∞,

$$\|f\_n - f\| \to 0 \text{ in } \mathcal{H} \quad \text{and} \quad \|Lf\_n - \mathbf{g}\| \to 0 \quad \text{in} \quad \mathcal{G} \Rightarrow \mathbf{g} = \mathbf{L}\mathbf{f} \dots$$

**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�G! is a *sesqui-linear form* (or *sesquilinear function*) if for *f*, *h*∈ H, *g*, *k*∈G and *α*, *β* ∈ ,

$$\Phi(\text{i})\,\Phi(af+\beta h,\text{g}) = a\Phi(f,\text{g}) + \beta\Phi(h,\text{g})\tag{15}$$

$$\Phi(\text{ii})\,\Phi(f,\text{ag}+\beta k) = \overline{a}\Phi(f,\text{g}) + \beta\Phi(f,k) \tag{16}$$

are satisfied where H and G are Hilbert spaces. **Remark.** If *L*∈*B*ð Þ H, G , then the sesqui-linear form Φ defined by

$$
\Phi(f, \mathbf{g}) = \langle Lf, \mathbf{g} \rangle\_G \tag{17}
$$

is bounded in the sense that

$$|\Phi(f, \mathbf{g})| \le \lambda \|f\|\_{\mathcal{H}} \|\mathbf{g}\|\_{\mathcal{G}} \text{ for } f \in \mathcal{H}, \text{ g} \in \mathcal{G},\tag{18}$$

where *λ*≥ ∥*L*∥*:*

**Remark.** If a sesqui-linear form Φ satisfies the condition (18), then for *f* ∈ H, the linear functional

$$\mathfrak{g} \mapsto \overline{\Phi(f, \mathfrak{g})}$$

is continuous on G*:* If we apply the Riesz theorem, then there exists uniquely *f* 0 ∈G satisfying

$$\|f'\|\_{\mathcal{G}} \le \lambda \|f\|\_{\mathcal{H}} \text{ and } \Phi(f, \mathcal{g}) = \langle f', \mathcal{g} \rangle\_{\mathcal{G}} \text{ for } \mathcal{g} \in \mathcal{G}.$$

Hence *f*↦*f* <sup>0</sup> becomes linear, and as a result we obtain

$$
\Phi(f, \mathfrak{g}) = \langle f', \mathfrak{g} \rangle\_{\mathcal{G}} = \langle Lf, \mathfrak{g} \rangle\_{\mathcal{G}}.
$$

**Adjoint operator.** If *<sup>L</sup>*∈*B*ð Þ <sup>H</sup>, <sup>G</sup> , then the unique operator *<sup>L</sup>*<sup>∗</sup> <sup>∈</sup>*B*ð Þ <sup>G</sup>, <sup>H</sup> satisfying

$$\Phi(f, \mathfrak{g}) = \langle f, L^\* \mathfrak{g} \rangle\_{\mathfrak{H}} \text{ for} \\ f \in \mathfrak{H}, \mathfrak{g} \in \mathcal{G} \tag{19}$$

**Note.** If *L*∈*B*ð Þ H , the quadratic form *φ* on H is defined by

and it is bounded

where *λ*≥ ∥*L*∥*:*

Φð Þ¼ *f*, *g*

for all *f*, *g* ∈ H*:*

(or *positive definite*) if

*positive definite*).

following sense

Hermitian.

quadratic form *φ* by the equation

said to be *self-adjoint* if *<sup>L</sup>* <sup>¼</sup> *<sup>L</sup>*<sup>∗</sup> .

1

*A Survey on Hilbert Spaces and Reproducing Kernels DOI: http://dx.doi.org/10.5772/intechopen.91479*

given in (27) for the related quadratic form

*φ*ð Þ¼ *f* h i *Lf*, *f* for *f* ∈ H, (26)

<sup>∣</sup>*φ*ð Þ*<sup>f</sup>* ∣ ≤*λ*<sup>∥</sup> *<sup>f</sup>* <sup>∥</sup><sup>2</sup> for *<sup>f</sup>* <sup>∈</sup> <sup>H</sup>, (27)

**Remark.** The sesqui-linear form Φ associated with *L* can be recovered from the

**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

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

j j h i *Lf*, *g*

h i *Kf*, *<sup>g</sup>* <sup>G</sup> 2

if and only if the continuous linear operator

on the direct sum Hilbert space H ⊕ G with

is positive definite.

**69**

**Remark**. If *L* is self-adjoint, then the norm of *L* coincides with the minimum of *λ*

**Positive definite operator.** A self-adjoint operator *L*∈*B*ð Þ H is said to be *positive*

h i *Lf*, *f* ≥0 for all *f* ∈ H*:*

If h i *Lf*, *f* ¼ 0 only when *f* ¼ 0, then *L* is said to be *strictly positive* (or, *strictly*

**Note.** For any positive operator *L*∈*B*ð Þ H , the Schwarz inequality holds in the

**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

> *L K* <sup>∗</sup> *K M*

*<sup>f</sup>* <sup>⊕</sup> *<sup>g</sup>*<sup>↦</sup> *Lf* <sup>þ</sup> *<sup>K</sup>* <sup>∗</sup> ð Þ*<sup>g</sup>* <sup>⊕</sup> ð Þ *Kf* <sup>þ</sup> *Mg*

<sup>4</sup> f g *<sup>φ</sup>*ð Þ� *<sup>f</sup>* <sup>þ</sup> *<sup>g</sup> <sup>φ</sup>*ð Þ *<sup>f</sup>* � *<sup>g</sup>* <sup>þ</sup> f g *<sup>φ</sup>*ð Þ� *<sup>f</sup>* <sup>þ</sup> *ig <sup>φ</sup>*ð Þ *<sup>f</sup>* � *ig* (28)

∥*L*∥ ¼ supf g jh ij *Lf*, *f* : ∥ *f* ∥ ≤1 *:* (29)

<sup>2</sup> <sup>≤</sup>h i *Lf*, *<sup>f</sup>* � h i *Lg*, *<sup>g</sup> :* (30)

<sup>≤</sup>h i *Lf*, *<sup>f</sup>* Hh i *Mg*, *<sup>g</sup>* <sup>G</sup> for *<sup>f</sup>* <sup>∈</sup> <sup>H</sup>, <sup>∈</sup><sup>G</sup> (31)

is called the *adjoint* of *L*.

**Remark.** By the definitions of *L* and *L*<sup>∗</sup> , it follows that

$$
\langle \mathsf{L}f, \mathsf{g} \rangle\_{\mathcal{G}} = \langle f, L^\* \mathsf{g} \rangle\_{\mathcal{H}} \text{ for} \\
f \in \mathsf{H}, \,\, \mathbf{g} \in \mathcal{G}. \tag{20}
$$

**Isometric property.** The adjoint operation is isometric if

$$\|\|L\|\| = \|L^\*\|\|\text{ is satisfied.}\tag{21}$$

**Remark.** Let H, G, and K be Hilbert spaces and *K* ∈*B*ð Þ G, K and *L*∈*B*ð Þ H, G be given. Then

$$\text{KL} \in \mathcal{B}(\mathcal{H}, \mathcal{K}) \text{ and } (\text{KL})^\* = L^\* K^\* \tag{22}$$

$$\text{Ker}(L) = (\text{Ran}(L^\*))^\perp \text{ and } (\text{Ker}(L))^\perp = \text{Clos}\{\text{Ran}(L)^\*\}\tag{23}$$

where Kerð Þ *L* is the kernel of *L* and Ranð Þ *L* is the range of *L*.

**Theorem 12.** If *L*, *M* ∈ *B*ð Þ H, G , then the following statements are mutually equivalent.


$$\|\mathbf{M}^\*\mathbf{g}\| \le \lambda \|\mathbf{L}^\*\mathbf{g}\| \text{ for } \mathbf{g} \in \mathcal{G}.$$

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

$$
\rho: \mathcal{H} \to \mathbb{C}
$$

is a *quadratic form* if for all *f* ∈ H and *ζ* ∈ ,

$$\varrho(\zeta f) = |\zeta|^2 \varrho(f) \tag{24}$$

and

$$
\rho(f+\mathbf{g}) + \rho(f-\mathbf{g}) = 2\{\rho(f) + \rho(\mathbf{g})\}\tag{25}
$$

are satisfied.

**Note.** If *L*∈*B*ð Þ H , the quadratic form *φ* on H is defined by

$$
\varphi(f) = \langle Lf, f \rangle \text{ for} f \in \mathcal{H}, \tag{26}
$$

and it is bounded

∥*f* 0

is called the *adjoint* of *L*.

i. Ranð Þ *M* ⊆Ranð Þ *L* .

Hence *f*↦*f*

*Functional Calculus*

satisfying

given. Then

equivalent.

and

**68**

are satisfied.

∥<sup>G</sup> ≤ *λ*∥ *f* ∥<sup>H</sup> and Φð Þ¼ *f*, *g f*

<sup>0</sup> becomes linear, and as a result we obtain

0 , *g*

**Adjoint operator.** If *<sup>L</sup>*∈*B*ð Þ <sup>H</sup>, <sup>G</sup> , then the unique operator *<sup>L</sup>*<sup>∗</sup> <sup>∈</sup>*B*ð Þ <sup>G</sup>, <sup>H</sup>

**Remark.** Let H, G, and K be Hilbert spaces and *K* ∈*B*ð Þ G, K and *L*∈*B*ð Þ H, G be

**Theorem 12.** If *L*, *M* ∈ *B*ð Þ H, G , then the following statements are mutually

<sup>∥</sup>*M*<sup>∗</sup> *<sup>g</sup>*∥ ≤ *<sup>λ</sup>*∥*L*<sup>∗</sup> *<sup>g</sup>*<sup>∥</sup> for *<sup>g</sup>* <sup>∈</sup>G*:*

*φ* : H !

*φ ζ*ð Þ¼ *<sup>f</sup>* j j *<sup>ζ</sup>* <sup>2</sup>

Φð Þ¼ *f*, *g f*

**Remark.** By the definitions of *L* and *L*<sup>∗</sup> , it follows that

**Isometric property.** The adjoint operation is isometric if

where Kerð Þ *L* is the kernel of *L* and Ranð Þ *L* is the range of *L*.

ii. There exists *K* ∈*B*ð Þ H such that *M* ¼ *LK:*

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

is a *quadratic form* if for all *f* ∈ H and *ζ* ∈ ,

iii. There exists 0≤*λ*< ∞ such that

0 , *g*

<sup>G</sup> <sup>¼</sup> h i *Lf*, *<sup>g</sup>* <sup>G</sup>*:*

<sup>Φ</sup>ð Þ¼ *<sup>f</sup>*, *<sup>g</sup> <sup>f</sup>*, *<sup>L</sup>*<sup>∗</sup> h i*<sup>g</sup>* <sup>H</sup> for *<sup>f</sup>* <sup>∈</sup> <sup>H</sup>, *<sup>g</sup>* <sup>∈</sup><sup>G</sup> (19)

h i *Lf*, *<sup>g</sup>* <sup>G</sup> <sup>¼</sup> *<sup>f</sup>*, *<sup>L</sup>*<sup>∗</sup> h i*<sup>g</sup>* <sup>H</sup> for *<sup>f</sup>* <sup>∈</sup> <sup>H</sup>, *<sup>g</sup>* <sup>∈</sup>G*:* (20)

<sup>∥</sup>*L*<sup>∥</sup> <sup>¼</sup> <sup>∥</sup>*L*<sup>∗</sup> <sup>∥</sup> is satisfied*:* (21)

*KL* <sup>∈</sup>*B*ð Þ <sup>H</sup>, <sup>K</sup> and ð Þ *KL* <sup>∗</sup> <sup>¼</sup> *<sup>L</sup>*∗*<sup>K</sup>* <sup>∗</sup> (22)

*φ*ð Þ*f* (24)

*φ*ð Þþ *f* þ *g φ*ð Þ¼ *f* � *g* 2f g *φ*ð Þþ *f φ*ð Þ*g* (25)

Kerð Þ¼ *<sup>L</sup>* Ran *<sup>L</sup>*<sup>∗</sup> ð Þ ð Þ <sup>⊥</sup> and Ker ð Þ ð Þ *<sup>L</sup>* <sup>⊥</sup> <sup>¼</sup> Clos Ranð Þ *<sup>L</sup>* <sup>∗</sup> f g (23)

<sup>G</sup> for *<sup>g</sup>* <sup>∈</sup>G*:*

$$|\varphi(f)| \le \lambda \|f\|^2 \text{ for} f \in \mathcal{H},\tag{27}$$

where *λ*≥ ∥*L*∥*:*

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

$$\Phi(f, \mathbf{g}) = \frac{1}{4} \{ \rho(f + \mathbf{g}) - \rho(f - \mathbf{g}) \} + \{ \rho(f + \mathbf{i}\mathbf{g}) - \rho(f - \mathbf{i}\mathbf{g}) \}\tag{28}$$

for all *f*, *g* ∈ H*:*

**Self-adjoint operator.** A continuous linear operator *L* on a Hilbert space H is said to be *self-adjoint* if *<sup>L</sup>* <sup>¼</sup> *<sup>L</sup>*<sup>∗</sup> .

**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

$$\|L\| = \sup\{ |\langle Lf, f\rangle| \colon \|f\| \le 1 \}.\tag{29}$$

**Positive definite operator.** A self-adjoint operator *L*∈*B*ð Þ H is said to be *positive* (or *positive definite*) if

$$
\langle Lf, f \rangle \ge 0 \quad \text{for all } f \in \mathcal{H}.
$$

If h i *Lf*, *f* ¼ 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*ð Þ H , the Schwarz inequality holds in the following sense

$$\left| \left< \mathcal{L}f, \mathbf{g} \right> \right|^2 \leq \left< \mathcal{L}f, f \right> \cdot \left< \mathcal{L}\mathbf{g}, \mathbf{g} \right>. \tag{30}$$

**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

$$\left| \langle \mathsf{Kf}, \mathsf{g} \rangle\_{\mathcal{G}} \right|^{2} \leq \langle \mathsf{L}f, f \rangle\_{\mathcal{H}} \langle \mathsf{Mg}, \mathsf{g} \rangle\_{\mathcal{G}} \text{ for } f \in \mathsf{H}, \mathsf{e} \in \mathcal{G} \tag{31}$$

if and only if the continuous linear operator

$$
\begin{bmatrix} L & K^\* \\ K & M \end{bmatrix}
$$

on the direct sum Hilbert space H ⊕ G with

$$f \oplus \mathbf{g} \mapsto (Lf + K^\*\mathbf{g}) \oplus (\mathbf{K}f + \mathbf{M}\mathbf{g}),$$

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 *L*1*=*<sup>2</sup> , such that *<sup>L</sup>*1*=*<sup>2</sup> <sup>2</sup> <sup>¼</sup> *<sup>L</sup>:*

**Modulus operator.** The square root of the positive definite operator *L*<sup>∗</sup> *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

$$\|\mathcal{U}f\|\_{\mathcal{J}} = \|f\|\_{\mathcal{H}} \text{ for} f \in \mathcal{H} \tag{32}$$

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

<sup>∥</sup>*Kx*<sup>∥</sup> <sup>¼</sup> h i *Kx*,*Kx* <sup>1</sup>*=*<sup>2</sup> <sup>¼</sup> *K x*ð Þ , *<sup>x</sup>*

**Note.** Observe that the subset f g *Kx <sup>x</sup>*∈*<sup>X</sup>* is total in H, that is, its closed linear span coincides with H. This follows from the fact that, if *f* ∈ H and *f*⊥*Kx* for all *x*∈ *X*,

*f x*ð Þ¼ h i *f*,*Kx* ¼ 0 for all *x*∈*X*,

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

**Theorem 18.** If a Hilbert space H of functions on a set *X* admits a reproducing

**Theorem 19.** There exists a reproducing kernel *K* for H for a Hilbert space H of

**Hermitian and positive definite kernel.** Let *X* be an arbitrary set and *K* be a kernel on *X*, that is, *K* : *X* � *X* ! . The kernel *K* is called *Hermitian* if for any finite

> , *yi* � �<sup>∈</sup> *:*

, *yi* � �≥0*:*

ϵ*y*ϵ*xK y* ð Þ , *x* ≥0*:* (38)

functions on *X*, if and only if for all *x*∈*X*, the linear functional H∍*f*↦*f x*ð Þ of

ϵ*j*ϵ*iK yj*

It is called *positive definite*, if for any complex numbers ϵ1, … , ϵ*n*, we have

ϵ*j*ϵ*iK yj*

**Note.** From the previous inequality, it follows that for any finitely supported

**Theorem 20.** The reproducing kernel *K* of a reproducing kernel Hilbert space H

**Properties of RKHS.** Given a reproducing kernel Hilbert space H and its kernel

� �, for *x*, *y*∈*X:* (36)

*:* (37)

1*=*2

*Kx*ð Þ¼ *y K y* ð Þ¼ , *x Ky*,*Kx*

and hence *<sup>f</sup>* is the 0 element in <sup>H</sup>. As a result, 0f g<sup>⊥</sup> ¼ H.

X*n i*, *j*¼1

X*n i*, *j*¼1

X *x*, *y*∈*X*

<sup>2</sup> <sup>≤</sup> *K y* ð Þ , *<sup>y</sup> K x*ð Þ , *<sup>x</sup>* (Schwarz inequality).

kernel *K*, then this reproducing kernel *K* is unique.

� �⊆*X*, we have

family of complex numbers f g ϵ*<sup>x</sup> <sup>x</sup>*∈*<sup>X</sup>*, we have

*K y* ð Þ , *x* on *X*, then for all *x*, *y*∈ *X*, we have

i. *K y* ð Þ , *y* ≥0*:*

iii. j j *K y* ð Þ , *x*

**71**

ii. *K y* ð Þ¼ , *x K x*ð Þ , *y :*

is a positive definite matrix in the sense of E.H. Moore.

Then, for any *x*∈*X*, we obtain

*A Survey on Hilbert Spaces and Reproducing Kernels DOI: http://dx.doi.org/10.5772/intechopen.91479*

reproducing kernel *K* of H.

evaluation at *x* is bounded on H*:*

set of points *y*1, … , *yn*

we get

then

is satisfied, that is, it preserves the norm.

**Note.** Eq. (32) implies that a continuous linear operator *U* is isometric if and only if *<sup>U</sup>* <sup>∗</sup> *<sup>U</sup>* <sup>¼</sup> *<sup>I</sup>*H; in other words,

$$
\langle \mathsf{U}f, \mathsf{Ug} \rangle\_{\mathfrak{G}} = \langle f, \mathsf{g} \rangle\_{\mathfrak{H}} \text{ for} \\
f, \mathfrak{g} \in \mathsf{H},
\tag{33}
$$

that is, *U* preserves the inner product.

**Unitary operator.** A surjective isometry linear operator *U* : H!H is called a *unitary* (*operator*).

**Note.** Observe that if *<sup>U</sup>* <sup>∈</sup>*B H*ð Þ is a unitary operator, then *<sup>U</sup>* <sup>∗</sup> <sup>¼</sup> *<sup>U</sup>*�<sup>1</sup> *:*

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

$$f \in (\text{Ker} U)^\perp = \text{Ran}(U^\*) \Rightarrow \|\| Uf \| = \|f\|.$$

The spaces Ker ð Þ *<sup>U</sup>* <sup>⊥</sup> and *Ran U*ð Þ are called the *initial space* of *<sup>U</sup>* and the *final space* of *U*, respectively.

**Note.** If *U* is a partial isometry, then its adjoint *U* <sup>∗</sup> is also a partial isometry.

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

$$L = U\tilde{L},\tag{34}$$

where *L*~ is a positive definite operator and *U* is a partial isometry with initial space the closure of Ran *L*~ .

#### **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 *<sup>X</sup>* with the inner product h i *<sup>f</sup>*, *<sup>g</sup>* and norm <sup>∥</sup> *<sup>f</sup>* <sup>∥</sup> <sup>¼</sup> h i *<sup>f</sup>*, *<sup>f</sup>* <sup>1</sup>*=*<sup>2</sup> for *<sup>f</sup>* and *<sup>g</sup>* <sup>∈</sup> <sup>H</sup>. Then the complex valued function *K y* ð Þ , *x* of *y* and *x* in *X* is called a *reproducing kernel of* H if

i. For all *x*∈*X*, it follows that *Kx*ðÞ¼ � *K*ð Þ �, *x* ∈ H,

ii. For all *x*∈*X* and all *f* ∈ H,

$$f(\mathbf{x}) = \langle f, K\_{\mathbf{x}} \rangle,\tag{35}$$

are satisfied.

*A Survey on Hilbert Spaces and Reproducing Kernels DOI: http://dx.doi.org/10.5772/intechopen.91479*

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

$$K\_x(y) = K(y, \mathfrak{x}) = \langle K\_\mathcal{V}, K\_\mathcal{x} \rangle, \text{ for } \mathfrak{x}, \mathfrak{y} \in \mathcal{X}. \tag{36}$$

Then, for any *x*∈*X*, we obtain

**Theorem 15.** Let *L* be a continuous positive definite operator. Then there exists a

**Modulus operator.** The square root of the positive definite operator *L*<sup>∗</sup> *L* is

**Isometry.** A linear operator *U* between Hilbert spaces H and G is called *isometric*

**Note.** Eq. (32) implies that a continuous linear operator *U* is isometric if and

**Unitary operator.** A surjective isometry linear operator *U* : H!H is called a

**Partial isometry.** A continuous linear operator *U* between Hilbert spaces H and

*<sup>f</sup>* <sup>∈</sup>ð Þ Ker*<sup>U</sup>* <sup>⊥</sup> <sup>¼</sup> Ran *<sup>U</sup>* <sup>∗</sup> ð Þ) <sup>∥</sup>*Uf* <sup>∥</sup> <sup>¼</sup> <sup>∥</sup> *<sup>f</sup>* <sup>∥</sup>*:*

The spaces Ker ð Þ *<sup>U</sup>* <sup>⊥</sup> and *Ran U*ð Þ are called the *initial space* of *<sup>U</sup>* and the *final*

**Note.** If *U* is a partial isometry, then its adjoint *U* <sup>∗</sup> is also a partial isometry. **Theorem 17.** Every continuous linear operator *L* on H admits a unique

where *L*~ is a positive definite operator and *U* is a partial isometry with initial

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

**Reproducing kernels.** Let H be a Hilbert space of functions on a nonempty set *<sup>X</sup>* with the inner product h i *<sup>f</sup>*, *<sup>g</sup>* and norm <sup>∥</sup> *<sup>f</sup>* <sup>∥</sup> <sup>¼</sup> h i *<sup>f</sup>*, *<sup>f</sup>* <sup>1</sup>*=*<sup>2</sup> for *<sup>f</sup>* and *<sup>g</sup>* <sup>∈</sup> <sup>H</sup>. Then the complex valued function *K y* ð Þ , *x* of *y* and *x* in *X* is called a *reproducing kernel of* H if

**Note.** Observe that if *<sup>U</sup>* <sup>∈</sup>*B H*ð Þ is a unitary operator, then *<sup>U</sup>* <sup>∗</sup> <sup>¼</sup> *<sup>U</sup>*�<sup>1</sup>

∥*Uf* ∥<sup>G</sup> ¼ ∥ *f* ∥<sup>H</sup> for *f* ∈ H (32)

h i *Uf*, *Ug* <sup>G</sup> <sup>¼</sup> h i *<sup>f</sup>*, *<sup>g</sup>* <sup>H</sup> for *<sup>f</sup>*, *<sup>g</sup>* <sup>∈</sup> <sup>H</sup>, (33)

*<sup>L</sup>* <sup>¼</sup> *UL*~, (34)

*f x*ð Þ¼ h i *f*, *Kx* , (35)

, such that

*:*

unique positive definite operator called the *square root* of *L*, denoted by *L*1*=*<sup>2</sup>

called the *modulus* (*operator*) of *L* if *L* is a continuous linear operator.

is satisfied, that is, it preserves the norm.

that is, *U* preserves the inner product.

only if *<sup>U</sup>* <sup>∗</sup> *<sup>U</sup>* <sup>¼</sup> *<sup>I</sup>*H; in other words,

*<sup>L</sup>*1*=*<sup>2</sup> <sup>2</sup> <sup>¼</sup> *<sup>L</sup>:*

*Functional Calculus*

or an *isometry* if

*unitary* (*operator*).

G is called a *partial isometry* if

*space* of *U*, respectively.

space the closure of Ran *L*~

.

i. For all *x*∈*X*, it follows that *Kx*ðÞ¼ � *K*ð Þ �, *x* ∈ H,

**3. Reproducing kernels and RKHS**

**3.1 Definition and fundamental properties**

ii. For all *x*∈*X* and all *f* ∈ H,

are satisfied.

**70**

decomposition

$$\|K\_{\mathbf{x}}\| = \langle K\_{\mathbf{x}}, K\_{\mathbf{x}} \rangle^{1/2} = K(\mathbf{x}, \mathbf{x})^{1/2}. \tag{37}$$

**Note.** Observe that the subset f g *Kx <sup>x</sup>*∈*<sup>X</sup>* is total in H, that is, its closed linear span coincides with H. This follows from the fact that, if *f* ∈ H and *f*⊥*Kx* for all *x*∈ *X*, then

$$f(\mathfrak{x}) = \langle f, K\_{\mathfrak{x}} \rangle = \mathbf{0} \text{ for all } \mathfrak{x} \in \mathbf{X},$$

and hence *<sup>f</sup>* is the 0 element in <sup>H</sup>. As a result, 0f g<sup>⊥</sup> ¼ 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 *x*∈*X*, the linear functional H∍*f*↦*f x*ð Þ 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* � *X* ! . The kernel *K* is called *Hermitian* if for any finite set of points *y*1, … , *yn* � �⊆*X*, we have

$$\sum\_{i,j=1}^n \overline{\mathbf{e}}\_j \mathbf{e}\_i K\left(\boldsymbol{\nu}\_j, \boldsymbol{\nu}\_i\right) \in \mathbb{R}\,.$$

It is called *positive definite*, if for any complex numbers ϵ1, … , ϵ*n*, we have

$$\sum\_{i,j=1}^n \overline{\epsilon}\_j \mathbf{e}\_i K\left(\boldsymbol{\nu}\_j, \boldsymbol{\nu}\_i\right) \ge \mathbf{0}.$$

**Note.** From the previous inequality, it follows that for any finitely supported family of complex numbers f g ϵ*<sup>x</sup> <sup>x</sup>*∈*<sup>X</sup>*, we have

$$\sum\_{\mathbf{x}, \mathbf{y} \in X} \overline{\epsilon}\_{\mathbf{y}} \epsilon\_{\mathbf{x}} K(\mathbf{y}, \mathbf{x}) \ge \mathbf{0}. \tag{38}$$

**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 *K y* ð Þ , *x* on *X*, then for all *x*, *y*∈ *X*, we have

$$\text{i. } K(\mathcal{y}, \mathcal{y}) \ge 0.$$

$$\text{ii. } K(\mathcal{y}, \mathfrak{x}) = \overline{K(\mathfrak{x}, \mathfrak{y})}.$$

iii. j j *K y* ð Þ , *x* <sup>2</sup> <sup>≤</sup> *K y* ð Þ , *<sup>y</sup> K x*ð Þ , *<sup>x</sup>* (Schwarz inequality). iv. Let *x*<sup>0</sup> ∈*X*. Then the following statements are equivalent:

$$\begin{array}{l} \text{a. } K(\mathfrak{x}\_{0}, \mathfrak{x}\_{0}) = \mathbf{0}. \\\\ \text{b. } K(\mathfrak{y}, \mathfrak{x}\_{0}) = \mathbf{0} \text{ for all } \mathfrak{y} \in \mathcal{X}. \\\\ \text{c. } f(\mathfrak{x}\_{0}) = \mathbf{0} \text{ for all } f \in \mathcal{H}. \end{array}$$

**Theorem 21.** For any positive definite kernel *K* on *X*, there exists a unique Hilbert space H*<sup>K</sup>* of functions on *X* with reproducing kernel *K*.

**Theorem 22.** Every sequence of functions *f <sup>n</sup>* � � *<sup>n</sup>* <sup>≥</sup><sup>1</sup> that converges strongly to a function *f* in H*K*ð Þ *X* converges also in the pointwise sense, i.e., for any point *x*∈ *X*,

$$\lim\_{n \to \infty} f\_n(\mathbf{x}) = f(\mathbf{x}).$$

In addition, this convergence is uniform on every subset of *X* on which *x*↦*K x*ð Þ , *x* is bounded.

**Theorem 23.** A complex valued function *g* on *X* belongs to the reproducing kernel Hilbert space H*K*ð Þ *X* if and only if there exists 0 ≤*λ*< ∞ such that,

$$\left[\underline{\mathbf{g}}(\boldsymbol{\mathcal{Y}})\overline{\underline{\mathbf{g}}(\boldsymbol{\omega})}\right] \leq \lambda^2 \left[\mathcal{K}(\boldsymbol{\mathcal{Y}}, \boldsymbol{\omega})\right] \text{ on } X. \tag{39}$$

where M is the orthogonal complement of Kerð Þ *T* , *P*<sup>M</sup> is the orthogonal

set *X*. Then there exists a Hilbert space H and a function *φ* : *X* ! H such that

*K y* ð Þ¼ , *x φx*, *φ<sup>y</sup>*

H*<sup>K</sup>*ð Þ <sup>0</sup> ð Þ¼f *X*<sup>0</sup> *f*

*<sup>K</sup>*ð Þ<sup>2</sup> ð Þ *<sup>y</sup>*, *<sup>x</sup>* , and *K y* ð Þ , *<sup>x</sup>* , respectively, and let *<sup>K</sup>* <sup>¼</sup> *<sup>K</sup>*ð Þ<sup>1</sup> <sup>þ</sup> *<sup>K</sup>*ð Þ<sup>2</sup> *:* Then

and for *f* ∈ H*<sup>K</sup>*ð Þ<sup>1</sup> ð Þ *X* and *g* ∈ H*<sup>K</sup>*ð Þ<sup>2</sup> ð Þ *X* , it follows that

In addition the intersection Hilbert space is a RKHS. **Theorem 29.** The reproducing kernel of the space

∥*h*∥*<sup>K</sup>*ð Þ <sup>0</sup> ¼ min ∥ *f* ∥*<sup>K</sup>* : *f*j

**Kolmogorov decomposition.** Let*K y* ð Þ , *x* be a positive definite kernel on an abstract

**Theorem 27.** Let *K*ð Þ <sup>0</sup> be the restriction of the positive definite kernel *K* to a nonempty subset *X*<sup>0</sup> of *X* and let H*K*ð Þ <sup>0</sup> ð Þ *X* and H*K*ð Þ *X* be the RKHS corresponding to

> � �

*<sup>X</sup>*<sup>0</sup> ¼ *h*

**Remark.** If *<sup>K</sup>*ð Þ<sup>1</sup> ð Þ *<sup>y</sup>*, *<sup>x</sup>* and *<sup>K</sup>*ð Þ<sup>2</sup> ð Þ *<sup>y</sup>*, *<sup>x</sup>* are two positive definite kernels, then

*K y* ð Þ¼ , *<sup>x</sup> <sup>K</sup>*ð Þ<sup>1</sup> ð Þþ *<sup>y</sup>*, *<sup>x</sup> <sup>K</sup>*ð Þ<sup>2</sup> ð Þ *<sup>y</sup>*, *<sup>x</sup>*

**Remark.** Let <sup>H</sup>*<sup>K</sup>*ð Þ<sup>1</sup> , <sup>H</sup>*<sup>K</sup>*ð Þ<sup>2</sup> , and <sup>H</sup>*<sup>K</sup>* be RKHSs with reproducing kernels *<sup>K</sup>*ð Þ<sup>1</sup> ð Þ *<sup>y</sup>*, *<sup>x</sup>* ,

H*K*ð Þ¼H *X <sup>K</sup>*ð Þ<sup>1</sup> ð ÞþH *X <sup>K</sup>*ð Þ<sup>2</sup> ð Þ *X* ,

**Theorem 28.** The intersection H*<sup>K</sup>*ð Þ<sup>1</sup> ð Þ *X* ∩H*<sup>K</sup>*ð Þ<sup>2</sup> ð Þ *X* of Hilbert spaces H*<sup>K</sup>*ð Þ<sup>1</sup> ð Þ *X* and

H*K*ð Þ¼H *X <sup>K</sup>*ð Þ<sup>1</sup> ð Þ *X* ∩H*<sup>K</sup>*ð Þ<sup>2</sup> ð Þ *X*

where ½ � ϵ*<sup>x</sup>* , *η<sup>x</sup>* ½ �, *ζ<sup>x</sup>* ½ � are an arbitrary complex valued function on *X* with finite

*<sup>K</sup>*ð Þ<sup>1</sup> <sup>þ</sup> <sup>∥</sup> *<sup>f</sup>* <sup>∥</sup><sup>2</sup>

*<sup>η</sup>yηxK*ð Þ<sup>1</sup> ð Þþ *<sup>y</sup>*, *<sup>x</sup>* <sup>X</sup>

*<sup>K</sup>*ð Þ<sup>1</sup> <sup>þ</sup> <sup>∥</sup>*<sup>g</sup>* � *<sup>h</sup>*∥<sup>2</sup>

H*<sup>K</sup>*ð Þ<sup>2</sup> ð Þ *X* is again a Hilbert space of functions on *X* with respect to the norm

∥ *f* ∥<sup>2</sup> ≔ ∥ *f* ∥<sup>2</sup>

X *x*, *y*

¼ *η<sup>x</sup>* ½ �þ *ζ<sup>x</sup>* ½ �g,

D E for *<sup>x</sup>*, *<sup>y</sup>*<sup>∈</sup> *<sup>X</sup>:*

*<sup>X</sup>*<sup>0</sup> : *f* ∈ H*K*ð Þg *X* (41)

n o for all *<sup>h</sup>*<sup>∈</sup> <sup>H</sup>*<sup>K</sup>*ð Þ <sup>0</sup> ð Þ *<sup>X</sup>*<sup>0</sup> *:* (42)

*<sup>K</sup>*ð Þ<sup>2</sup> : *h*∈ H*<sup>K</sup>*ð Þ<sup>1</sup> ð Þ *X* ∩H*<sup>K</sup>*ð Þ<sup>2</sup> ð Þ *X*

n o*:* (43)

*<sup>K</sup>*ð Þ<sup>2</sup> *:*

*x*, *y*

*<sup>ζ</sup>yζxK*ð Þ<sup>2</sup> ð Þ *<sup>y</sup>*, *<sup>x</sup>* : ½ � *<sup>ε</sup><sup>x</sup>*

projection onto M, and ∥ � ∥*<sup>K</sup>* denotes the norm in H*K*ð Þ *X :*

*A Survey on Hilbert Spaces and Reproducing Kernels DOI: http://dx.doi.org/10.5772/intechopen.91479*

**3.2 Operations with RKHSs**

*K*ð Þ <sup>0</sup> and *K*, respectively. Then

is also a positive definite kernel.

*<sup>K</sup>* <sup>¼</sup> min <sup>∥</sup> *<sup>f</sup>* <sup>þ</sup> *<sup>h</sup>*∥<sup>2</sup>

is determined, as a quadratic form, by

*εyεxK y* ð Þ¼ , *x* inf f

and

<sup>∥</sup> *<sup>f</sup>* <sup>þ</sup> *<sup>g</sup>*∥<sup>2</sup>

X *x*, *y*

support.

**73**

∥*g*∥ coincides with the minimum of all such *λ*.

**Theorem 24.** If *<sup>K</sup>*ð Þ<sup>1</sup> ð Þ *<sup>y</sup>*, *<sup>x</sup>* and *<sup>K</sup>*ð Þ<sup>2</sup> ð Þ *<sup>y</sup>*, *<sup>x</sup>* are two positive definite kernels on *<sup>X</sup>*, then the following statements are mutually equivalent:

$$\text{i. } \mathcal{H}\_{K^{(1)}}(X) \subseteq \mathcal{H}\_{K^{(2)}}(X).$$

ii. There exists 0≤*λ*< ∞ such that

$$\left[K^{(1)}(\mathcal{Y},\mathcal{X})\right] \leq \lambda^2 \left[K^{(2)}(\mathcal{Y},\mathcal{X})\right].$$

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

$$K(\boldsymbol{\wp}, \boldsymbol{\varkappa}) = \left\langle \boldsymbol{\wp}\_{\boldsymbol{\varkappa}}, \boldsymbol{\varrho}\_{\boldsymbol{\jmath}} \right\rangle \text{ for } \boldsymbol{\varkappa}, \boldsymbol{\jmath} \in X. \tag{40}$$

**Theorem 25.** Let *φ* : *X*↦H be an arbitrary map and for *x*, *y*∈*X* let *K* be defined as

$$K(\boldsymbol{\jmath}, \boldsymbol{\varkappa}) = \left\langle \boldsymbol{\varrho}\_{\boldsymbol{\varkappa}}, \boldsymbol{\varrho}\_{\boldsymbol{\jmath}} \right\rangle.$$

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

$$(Tf)(\mathfrak{x}) = \langle f, \rho\_{\mathfrak{x}} \rangle \text{ for } \mathfrak{x} \in \mathcal{X}, f \in \mathcal{H}.$$

Then Ranð Þ *T* coincides with H*K*ð Þ *X* and

$$\|Tf\|\_{K} = \|P\_{\mathcal{M}}f\| \text{ for} f \in \mathcal{H},$$

*A Survey on Hilbert Spaces and Reproducing Kernels DOI: http://dx.doi.org/10.5772/intechopen.91479*

where M is the orthogonal complement of Kerð Þ *T* , *P*<sup>M</sup> is the orthogonal projection onto M, and ∥ � ∥*<sup>K</sup>* denotes the norm in H*K*ð Þ *X :*

**Kolmogorov decomposition.** Let*K y* ð Þ , *x* be a positive definite kernel on an abstract set *X*. Then there exists a Hilbert space H and a function *φ* : *X* ! H such that

$$K(\boldsymbol{\jmath}, \boldsymbol{\varkappa}) = \left\langle \boldsymbol{\varrho}\_{\boldsymbol{\varkappa}}, \boldsymbol{\varrho}\_{\boldsymbol{\jmath}} \right\rangle \text{ for } \boldsymbol{\varkappa}, \boldsymbol{\jmath} \in X.$$

#### **3.2 Operations with RKHSs**

**Theorem 27.** Let *K*ð Þ <sup>0</sup> be the restriction of the positive definite kernel *K* to a nonempty subset *X*<sup>0</sup> of *X* and let H*K*ð Þ <sup>0</sup> ð Þ *X* and H*K*ð Þ *X* be the RKHS corresponding to *K*ð Þ <sup>0</sup> and *K*, respectively. Then

$$\mathcal{H}\_{K^{(0)}}(X\_0) = \{ f \vert\_{X\_0} : f \in \mathcal{H}\_K(X) \} \tag{41}$$

and

iv. Let *x*<sup>0</sup> ∈*X*. Then the following statements are equivalent:

Hilbert space H*<sup>K</sup>* of functions on *X* with reproducing kernel *K*.

**Theorem 22.** Every sequence of functions *f <sup>n</sup>*

**Theorem 21.** For any positive definite kernel *K* on *X*, there exists a unique

function *f* in H*K*ð Þ *X* converges also in the pointwise sense, i.e., for any point *x*∈ *X*,

lim*<sup>n</sup>*!<sup>∞</sup> *<sup>f</sup> <sup>n</sup>*ð Þ¼ *<sup>x</sup> f x*ð Þ*:*

**Theorem 23.** A complex valued function *g* on *X* belongs to the reproducing

**Theorem 24.** If *<sup>K</sup>*ð Þ<sup>1</sup> ð Þ *<sup>y</sup>*, *<sup>x</sup>* and *<sup>K</sup>*ð Þ<sup>2</sup> ð Þ *<sup>y</sup>*, *<sup>x</sup>* are two positive definite kernels on *<sup>X</sup>*,

<sup>≤</sup> *<sup>λ</sup>*<sup>2</sup> *<sup>K</sup>*ð Þ<sup>2</sup> ð Þ *<sup>y</sup>*, *<sup>x</sup>* h i

D E

*:*

**Note.** For any map *φ* from a set *X* to a Hilbert space H, with the notation *x*↦*φx*,

D E

*K y* ð Þ¼ , *x φx*, *φ<sup>y</sup>*

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

ð Þ *Tf* ð Þ¼ *x f*, *φ<sup>x</sup>* h i for *x*∈*X*, *f* ∈ H*:*

∥*Tf* ∥*<sup>K</sup>* ¼ ∥*P*M*f* ∥ for *f* ∈ H,

**Theorem 25.** Let *φ* : *X*↦H be an arbitrary map and for *x*, *y*∈*X* let *K* be

*:*

for *x*, *y*∈ *X:* (40)

In addition, this convergence is uniform on every subset of *X* on which

kernel Hilbert space H*K*ð Þ *X* if and only if there exists 0 ≤*λ*< ∞ such that,

*g y*ð Þ*g x*ð Þ h i

*<sup>K</sup>*ð Þ<sup>1</sup> ð Þ *<sup>y</sup>*, *<sup>x</sup>* h i

*K y* ð Þ¼ , *x φx*, *φ<sup>y</sup>*

∥*g*∥ coincides with the minimum of all such *λ*.

then the following statements are mutually equivalent:

ii. There exists 0≤*λ*< ∞ such that

Then *K* is a positive definite kernel.

Then Ranð Þ *T* coincides with H*K*ð Þ *X* and

i. H*<sup>K</sup>*ð Þ<sup>1</sup> ð Þ *X* ⊆ H*<sup>K</sup>*ð Þ<sup>2</sup> ð Þ *X* .

a kernel *K* can be defined by

defined as

defined by

**72**

� �

*<sup>n</sup>* <sup>≥</sup><sup>1</sup> that converges strongly to a

<sup>≤</sup> *<sup>λ</sup>*<sup>2</sup> ½ � *K y* ð Þ , *<sup>x</sup>* on *<sup>X</sup>:* (39)

a. *K x*ð Þ¼ 0, *x*<sup>0</sup> 0.

*Functional Calculus*

*x*↦*K x*ð Þ , *x* is bounded.

b. *K y* ð Þ¼ , *x*<sup>0</sup> 0 for all *y*∈*X*.

c. *f x*ð Þ¼ <sup>0</sup> 0 for all *f* ∈ H*:*

$$\|h\|\_{K^{(0)}} = \min\left\{ \|f\|\_{K} : f|\_{X\_0} = h \right\} \text{ for all } h \in \mathcal{H}\_{K^{(0)}}(X\_0). \tag{42}$$

**Remark.** If *<sup>K</sup>*ð Þ<sup>1</sup> ð Þ *<sup>y</sup>*, *<sup>x</sup>* and *<sup>K</sup>*ð Þ<sup>2</sup> ð Þ *<sup>y</sup>*, *<sup>x</sup>* are two positive definite kernels, then

$$K(\boldsymbol{\jmath}, \boldsymbol{\varkappa}) = K^{(1)}(\boldsymbol{\jmath}, \boldsymbol{\varkappa}) + K^{(2)}(\boldsymbol{\jmath}, \boldsymbol{\varkappa})$$

is also a positive definite kernel.

**Remark.** Let <sup>H</sup>*<sup>K</sup>*ð Þ<sup>1</sup> , <sup>H</sup>*<sup>K</sup>*ð Þ<sup>2</sup> , and <sup>H</sup>*<sup>K</sup>* be RKHSs with reproducing kernels *<sup>K</sup>*ð Þ<sup>1</sup> ð Þ *<sup>y</sup>*, *<sup>x</sup>* , *<sup>K</sup>*ð Þ<sup>2</sup> ð Þ *<sup>y</sup>*, *<sup>x</sup>* , and *K y* ð Þ , *<sup>x</sup>* , respectively, and let *<sup>K</sup>* <sup>¼</sup> *<sup>K</sup>*ð Þ<sup>1</sup> <sup>þ</sup> *<sup>K</sup>*ð Þ<sup>2</sup> *:* Then

$$
\mathcal{H}\_K(X) = \mathcal{H}\_{K^{(1)}}(X) + \mathcal{H}\_{K^{(2)}}(X),
$$

and for *f* ∈ H*<sup>K</sup>*ð Þ<sup>1</sup> ð Þ *X* and *g* ∈ H*<sup>K</sup>*ð Þ<sup>2</sup> ð Þ *X* , it follows that

$$\|f+\mathbf{g}\|\_{K}^{2} = \min\left\{\|f+h\|\_{K^{(1)}}^{2} + \|\mathbf{g}-h\|\_{K^{(2)}}^{2} : h \in \mathcal{H}\_{K^{(1)}}(X) \cap \mathcal{H}\_{K^{(2)}}(X)\right\}.\tag{43}$$

**Theorem 28.** The intersection H*<sup>K</sup>*ð Þ<sup>1</sup> ð Þ *X* ∩H*<sup>K</sup>*ð Þ<sup>2</sup> ð Þ *X* of Hilbert spaces H*<sup>K</sup>*ð Þ<sup>1</sup> ð Þ *X* and H*<sup>K</sup>*ð Þ<sup>2</sup> ð Þ *X* is again a Hilbert space of functions on *X* with respect to the norm

$$\|\|f\|\|^2 := \|f\|\_{K^{(1)}}^2 + \|f\|\_{K^{(2)}}^2.$$

In addition the intersection Hilbert space is a RKHS. **Theorem 29.** The reproducing kernel of the space

$$\mathcal{H}\_{\mathbb{K}}(X) = \mathcal{H}\_{\mathbb{K}^{(1)}}(X) \cap \mathcal{H}\_{\mathbb{K}^{(2)}}(X)$$

is determined, as a quadratic form, by

$$\begin{aligned} \sum\_{\mathbf{x}, \mathbf{y}} \overline{\epsilon\_{\mathbf{y}}} \epsilon\_{\mathbf{x}} K(\mathbf{y}, \mathbf{x}) &= \inf \left\{ \sum\_{\mathbf{x}, \mathbf{y}} \overline{\eta\_{\mathbf{y}}} \eta\_{\mathbf{x}} K^{(1)}(\mathbf{y}, \mathbf{x}) + \sum\_{\mathbf{x}, \mathbf{y}} \overline{\zeta\_{\mathbf{y}}} \zeta\_{\mathbf{x}} K^{(2)}(\mathbf{y}, \mathbf{x}) : [\mathbf{e}\_{\mathbf{x}}] \right\} \\ &= [\eta\_{\mathbf{x}}] + [\zeta\_{\mathbf{x}}]), \end{aligned}$$

where ½ � ϵ*<sup>x</sup>* , *η<sup>x</sup>* ½ �, *ζ<sup>x</sup>* ½ � are an arbitrary complex valued function on *X* with finite support.

**Theorem 30.** The tensor product Hilbert space

$$\mathcal{H}\_{\boldsymbol{K}^{(1)}}(\boldsymbol{X}) \otimes \mathcal{H}\_{\boldsymbol{K}^{(2)}}(\boldsymbol{X})$$

**Green function.** A *Green function G w*ð Þ , *z* 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 *G w*ð Þ¼ , *z* 0 for all *w* ∈ ∂Ω, where Ω is a finitely connected

curves, and let *G w*ð Þ , *z* be the Green's function of Ω*:* Then the Bergman kernel

2 *π ∂*2 *G ∂w∂z*

normal to all *φ<sup>n</sup>* (*n* <0), that is, all Fourier coefficients of *f* with negative indices

*an* ¼ *f*, *φ<sup>n</sup>* h i*L*<sup>2</sup> and *bn* ¼ *g*, *φ<sup>n</sup>* h i*L*<sup>2</sup> ð Þ *n* ¼ 0, 1, … *:*

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

*anbn* for *f*, *g* ∈ *H*<sup>2</sup>

*<sup>B</sup>*ð Þ <sup>Ω</sup> ð Þ¼� *<sup>w</sup>*, *<sup>z</sup>*

h i *<sup>f</sup>*, *<sup>g</sup> <sup>L</sup>*<sup>2</sup> <sup>¼</sup> <sup>X</sup><sup>∞</sup>

<sup>1</sup>�*wz* for *<sup>w</sup>*, *<sup>z</sup>*<sup>∈</sup> is called the *Szegö kernel*.

(Hilbert type) *Hardy space* on and is denoted by *H*<sup>2</sup>

**Szegö kernel.** The kernel *<sup>S</sup>*ð Þ *<sup>ξ</sup>*, *<sup>z</sup>* <sup>≔</sup> <sup>1</sup>

explained in the books [7–9].

**Hardy space.** The closed linear span of *<sup>φ</sup><sup>n</sup>* f g : *<sup>n</sup>* <sup>¼</sup> 0, 1, … in *<sup>L</sup>*<sup>2</sup>

ð Þ belongs to the Hardy space *<sup>H</sup>*<sup>2</sup>

*n*¼0

**Theorem 33.** Let Ω be a finitely connected domain bounded by analytic Jordan

ð Þ *w*, *z* , *w* 6¼ *z:* (47)

ð Þ *:* Here *<sup>φ</sup>n*ð Þ¼ *<sup>ξ</sup> <sup>ξ</sup>n*.

<sup>1</sup>�*ξ<sup>z</sup>* for *<sup>ξ</sup>*<sup>∈</sup> , *<sup>z</sup>*<sup>∈</sup> , or its analytic extension

ð Þ if and only if it is ortho-

ð Þ , (48)

ðð Þ *T* is called the

domain of the complex plane.

*A Survey on Hilbert Spaces and Reproducing Kernels DOI: http://dx.doi.org/10.5772/intechopen.91479*

function is

**Remark.** *f* ∈ *L*<sup>2</sup>

vanish. Then we have

where

*S w*ð Þ , *<sup>z</sup>* <sup>≔</sup> <sup>1</sup>

**Notes**

~

**75**

is a RKHS on *X* � *X*.

**Theorem 31.** The RKHS <sup>H</sup>*K*ð Þ *<sup>X</sup>* of the kernel *K y* ð Þ¼ , *<sup>x</sup> <sup>K</sup>*ð Þ<sup>1</sup> ð Þ� *<sup>y</sup>*, *<sup>x</sup> <sup>K</sup>*ð Þ<sup>2</sup> ð Þ *<sup>y</sup>*, *<sup>x</sup>* consists of all functions *f* on *X* for which there are sequences *gn* � � *<sup>n</sup>*≥<sup>0</sup> of functions in H*K*ð Þ<sup>1</sup> ð Þ *X* and ð Þ *hn <sup>n</sup>*≥<sup>0</sup> of functions in H*K*ð Þ<sup>2</sup> ð Þ *X* so that

$$\sum\_{1}^{\infty} \left\| \mathbf{g}\_{n} \right\|\_{K^{(1)}}^{2} \left\| h\_{n} \right\|\_{K^{(2)}}^{2} < \infty, \quad \sum\_{1}^{\infty} \mathbf{g}\_{n}(\mathbf{x}) h\_{n}(\mathbf{x}) = f(\mathbf{x}), \ \mathbf{x} \in X,\tag{44}$$

and the norm is given by

$$\|f\|\_{K}^{2} = \min\left\{\sum\_{1}^{\infty} \left\|\mathbf{g}\_{n}\right\|\_{K^{(1)}}^{2} \left\|h\_{n}\right\|\_{K^{(2)}}^{2}\right\},$$

where the minimum is taken over the set of all sequences *gn* � � *<sup>n</sup>*≥<sup>0</sup> and ð Þ *hn* <sup>≥</sup><sup>0</sup> satisfying (44).

#### **3.3 Examples of RKHS. Bergman and Hardy spaces**

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

$$\int\_{\mathfrak{U}} \int\_{\mathfrak{U}} |f(x)|^2 dx dy < \infty, \qquad (x = \mathfrak{x} + i\mathfrak{y})^2$$

is satisfied is called the *Bergman space* on Ω and denoted by *A*<sup>2</sup> ð Þ Ω . **Remark.** *A*<sup>2</sup> ð Þ Ω is a *RKHS* with respect to the inner product

$$\langle f, \mathfrak{g} \rangle \equiv \langle f, \mathfrak{g} \rangle\_{\Omega} \coloneqq \int \int\_{\Omega} f(x) \overline{\mathfrak{g}(x)} d\mathfrak{x} d\mathfrak{y},$$

and its kernel is called the *Bergman kernel* on <sup>Ω</sup> and denoted by *<sup>B</sup>*ð Þ <sup>Ω</sup> ð Þ *<sup>w</sup>*, *<sup>z</sup> :*

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

$$B^{(\mathbb{D})}(w, z) = \frac{1}{\pi} \frac{1}{\left(1 - w\overline{z}\right)^2} \qquad \qquad \text{for } w, z \in \mathbb{D}. \tag{45}$$

**Bergman kernel of a simply connected domain.** The Bergman kernel of a simply connected domain Ωð Þ 6¼ is given by

$$B^{(\Omega)}(w, z) = \frac{1}{\pi} \frac{\rho'(w)\overline{\rho'(z)}}{\left(1 - \rho(w)\overline{\rho(z)}\right)^2} \text{ for } w, z \in \Omega,\tag{46}$$

where *φ* is any conformal mapping function from Ω onto *:*

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

**Jordan curve.** A *Jordan curve* is a continuous 1 � 1 image of f g j*ξ*j¼ 1 in .

*A Survey on Hilbert Spaces and Reproducing Kernels DOI: http://dx.doi.org/10.5772/intechopen.91479*

**Green function.** A *Green function G w*ð Þ , *z* 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 *G w*ð Þ¼ , *z* 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 *G w*ð Þ , *z* be the Green's function of Ω*:* Then the Bergman kernel function is

$$B^{(\Omega)}(w, z) = -\frac{2}{\pi} \frac{\partial^2 G}{\partial w \partial \overline{z}}(w, z), \ w \neq z. \tag{47}$$

**Hardy space.** The closed linear span of *<sup>φ</sup><sup>n</sup>* f g : *<sup>n</sup>* <sup>¼</sup> 0, 1, … in *<sup>L</sup>*<sup>2</sup> ðð Þ *T* is called the (Hilbert type) *Hardy space* on and is denoted by *H*<sup>2</sup> ð Þ *:* Here *<sup>φ</sup>n*ð Þ¼ *<sup>ξ</sup> <sup>ξ</sup>n*.

**Remark.** *f* ∈ *L*<sup>2</sup> ð Þ belongs to the Hardy space *<sup>H</sup>*<sup>2</sup> ð Þ if and only if it is orthonormal to all *φ<sup>n</sup>* (*n* <0), that is, all Fourier coefficients of *f* with negative indices vanish. Then we have

$$\langle f, \mathbf{g} \rangle\_{L^2} = \sum\_{n=0}^{\infty} a\_n \overline{b}\_n \text{ for} \\ f, \mathbf{g} \in H^2(\mathbb{T}), \tag{48}$$

where

**Theorem 30.** The tensor product Hilbert space

H*K*ð Þ<sup>1</sup> ð Þ *X* and ð Þ *hn <sup>n</sup>*≥<sup>0</sup> of functions in H*K*ð Þ<sup>2</sup> ð Þ *X* so that

∥ *f* ∥<sup>2</sup>

**3.3 Examples of RKHS. Bergman and Hardy spaces**

ð ð Ω j j *f z*ð Þ <sup>2</sup>

*<sup>B</sup>*ð Þ ð Þ¼ *<sup>w</sup>*, *<sup>z</sup>*

simply connected domain Ωð Þ 6¼ is given by

*<sup>B</sup>*ð Þ <sup>Ω</sup> ð Þ¼ *<sup>w</sup>*, *<sup>z</sup>*

*<sup>K</sup>*ð Þ<sup>1</sup> <sup>∥</sup>*hn*∥<sup>2</sup>

consists of all functions *f* on *X* for which there are sequences *gn*

*<sup>K</sup>*ð Þ<sup>2</sup> <sup>&</sup>lt; <sup>∞</sup>, <sup>X</sup><sup>∞</sup>

*<sup>K</sup>* <sup>¼</sup> min <sup>X</sup><sup>∞</sup>

where the minimum is taken over the set of all sequences *gn*

is a RKHS on *X* � *X*.

*Functional Calculus*

X∞ 1

satisfying (44).

**Remark.** *A*<sup>2</sup>

is given by

Bergman kernel of Ω*:*

**74**

∥*gn*∥<sup>2</sup>

and the norm is given by

H*K*ð Þ<sup>1</sup> ð Þ *X* ⊗ H*K*ð Þ<sup>2</sup> ð Þ *X*

**Theorem 31.** The RKHS <sup>H</sup>*K*ð Þ *<sup>X</sup>* of the kernel *K y* ð Þ¼ , *<sup>x</sup> <sup>K</sup>*ð Þ<sup>1</sup> ð Þ� *<sup>y</sup>*, *<sup>x</sup> <sup>K</sup>*ð Þ<sup>2</sup> ð Þ *<sup>y</sup>*, *<sup>x</sup>*

1

1

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

ð Þ Ω is a *RKHS* with respect to the inner product

and its kernel is called the *Bergman kernel* on <sup>Ω</sup> and denoted by *<sup>B</sup>*ð Þ <sup>Ω</sup> ð Þ *<sup>w</sup>*, *<sup>z</sup> :* **Bergman kernel for the unit disc.** The Bergman kernel for the open unit disc

> 1 ð Þ 1 � *wz*

> > *φ*0 ð Þ *w φ*<sup>0</sup> ð Þ*z*

**Bergman kernel of a simply connected domain.** The Bergman kernel of a

1 � *φ*ð Þ *w φ*ð Þ*z*

**Theorem 32.** A conformal mapping from Ω to can be recovered from the

**Jordan curve.** A *Jordan curve* is a continuous 1 � 1 image of f g j*ξ*j¼ 1 in .

ð ð Ω

is satisfied is called the *Bergman space* on Ω and denoted by *A*<sup>2</sup>

h i *f*, *g* � h i *f*, *g* <sup>Ω</sup> ≔

1 *π*

1 *π*

where *φ* is any conformal mapping function from Ω onto *:*

∥*gn*∥<sup>2</sup>

( )

*dxdy*< ∞, ð Þ *z* ¼ *x* þ *iy*

*f z*ð Þ*g z*ð Þ*dxdy*,

*<sup>K</sup>*ð Þ<sup>1</sup> <sup>∥</sup>*hn*∥<sup>2</sup>

*K*ð Þ<sup>2</sup>

,

� �

ð Þ Ω .

<sup>2</sup> for *w*, *z*∈ *:* (45)

� �<sup>2</sup> for *<sup>w</sup>*, *<sup>z</sup>*<sup>∈</sup> <sup>Ω</sup>, (46)

� �

*gn*ð Þ *x hn*ð Þ¼ *x f x*ð Þ, *x*∈*X*, (44)

*<sup>n</sup>*≥<sup>0</sup> of functions in

*<sup>n</sup>*≥<sup>0</sup> and ð Þ *hn* <sup>≥</sup><sup>0</sup>

$$\mathfrak{a}\_n = \langle f, \mathfrak{q}\_n \rangle\_{L^2} \text{ and } b\_n = \langle \mathfrak{g}, \mathfrak{q}\_n \rangle\_{L^2} \quad (n = 0, 1, \dots).$$

**Szegö kernel.** The kernel *<sup>S</sup>*ð Þ *<sup>ξ</sup>*, *<sup>z</sup>* <sup>≔</sup> <sup>1</sup> <sup>1</sup>�*ξ<sup>z</sup>* for *<sup>ξ</sup>*<sup>∈</sup> , *<sup>z</sup>*<sup>∈</sup> , or its analytic extension ~ *S w*ð Þ , *<sup>z</sup>* <sup>≔</sup> <sup>1</sup> <sup>1</sup>�*wz* for *<sup>w</sup>*, *<sup>z</sup>*<sup>∈</sup> is called the *Szegö kernel*.

### **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 [7–9].

*Functional Calculus*
