# Hilbert spaces

Around the turn of the nineteenth century into the twentieth century David Hilbert and others developed the type of vector space that later got Hilbert's name.

The Hilbert space is a particular vector space because it defines an inner product for every pair of its member vectors.

That inner product can take values of a number system for which every non-zero member owns a unique inverse. This requirement brands the number system as a division ring.

##### Bra's and ket's

Paul Dirac introduced a handy formulation for the inner product that applies a bra and a ket.

The bra ${\displaystyle \langle f|}$ is a covariant vector and the ket ${\displaystyle |g\rangle }$ is a contra-variant vector. The inner product acts as a metric.

For bra vectors hold

${\displaystyle \langle f|+\langle g|=\langle g|+\langle f|=\langle f+g|}$

(1 )

${\displaystyle (\langle f|+\langle g|)+\langle h|=\langle f|+(\langle g|+\langle h|)}$

(2 )

For ket vectors hold

${\displaystyle |f\rangle +|g\rangle =|g\rangle +|f\rangle =|f+g\rangle }$

(3 )

${\displaystyle (|f\rangle +|g\rangle )+|h\rangle =|f\rangle +(|g\rangle +|h\rangle )}$

(4 )

##### Inner product

For the inner product ${\displaystyle s=\langle f|g\rangle }$ holds that ${\displaystyle s}$ is a member of a division ring.

${\displaystyle \langle f|g\rangle {\overset {\underset {\mathrm {def} }{}}{=}}\langle g|f\rangle ^{*}}$

(5 )

For quaternionic numbers ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ hold

${\displaystyle \langle \alpha f|g\rangle =\langle g|\alpha f\rangle ^{*}=(\langle g|f\rangle \alpha )^{*}=\alpha ^{*}\langle f|g\rangle }$

(6 )

${\displaystyle \langle f|\beta g\rangle =\langle f|g\rangle \beta }$

(7 )

${\displaystyle \langle (\alpha +\beta )f|g\rangle =\langle \alpha f|g\rangle +\langle \beta f|g\rangle =\langle f|g\rangle \alpha +\langle f|g\rangle \beta }$

(8 )

Thus

${\displaystyle \langle \alpha f|=\alpha ^{*}\langle f|}$

(9 )

${\displaystyle |\alpha g\rangle =|g\rangle \alpha }$

(10 )

We made a choice. We could instead define ${\displaystyle \langle \alpha f|=\alpha \langle f|}$ and ${\displaystyle |\alpha g\rangle =|g\rangle \alpha ^{*}}$.

### Separable

In mathematics a topological space is called separable if it contains a countable dense subset; that is, there exists a sequence ${\displaystyle \{|x_{i}\rangle \}_{\infty }^{i=0}}$ of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

Its values on this countable dense subset determine every continuous function on the separable space ${\displaystyle {\mathfrak {H}}}$.

The Hilbert space ${\displaystyle {\mathfrak {H}}}$ is separable. That means that a countable row of elements ${\displaystyle \{|f_{n}\rangle \}}$ exists that spans the whole space.

If ${\displaystyle \langle f_{m}|f_{n}\rangle =\delta (m,n)}$ = [1 when ${\displaystyle m=n}$; 0 otherwise] then ${\displaystyle \{|f_{n}\rangle \}}$ forms an orthonormal base of the Hilbert space.

A ket base ${\displaystyle \{|k\rangle \}}$ of ${\displaystyle {\mathfrak {H}}}$ is a minimal set of ket vectors ${\displaystyle |k\rangle }$ that together span the Hilbert space ${\displaystyle {\mathfrak {H}}}$.

Any ket vector ${\displaystyle |f\rangle }$ in ${\displaystyle {\mathfrak {H}}}$ can be written as a linear combination of elements of ${\displaystyle \{|k\rangle \}}$.

${\displaystyle |f\rangle =\sum _{k}|k\rangle \langle k|f\rangle }$

(11 )

A bra base ${\displaystyle \{\langle b|\}}$ of ${\displaystyle {\mathfrak {H}}^{\dagger }}$ is a minimal set of bra vectors ${\displaystyle \langle b|}$ that together span the Hilbert space ${\displaystyle {\mathfrak {H}}^{\dagger }}$.

Any bra vector ${\displaystyle \langle f|}$ in ${\displaystyle {\mathfrak {H}}^{\dagger }}$ can be written as a linear combination of elements of ${\displaystyle \{\langle b|\}}$.

${\displaystyle \langle f|=\sum _{b}\langle f|b\rangle \langle b|}$

(12 )

Usually, a base selects vectors such that their norm equals 1. Such a base is called an orthonormal base.

#### Operators

Operators act on a subset of the elements of the Hilbert space.

##### Linear operators

An operator ${\displaystyle L}$ is linear when for all vectors ${\displaystyle |f\rangle }$ and ${\displaystyle |g\rangle }$ for which ${\displaystyle L}$ is defined and for all quaternionic numbers ${\displaystyle \alpha }$ and ${\displaystyle \beta }$

${\displaystyle |L\alpha f\rangle +|L\beta g\rangle =|Lf\rangle \alpha +|Lg\rangle \beta =L(|\alpha f\rangle +|\beta g\rangle )=L(|f\rangle \alpha +|g\rangle \beta )}$

(13 )

Operator ${\displaystyle B}$ is colinear when for all vectors ${\displaystyle |f\rangle }$ for which ${\displaystyle B}$ is defined and for all quaternionic numbers ${\displaystyle \alpha }$ there exists a quaternionic number ${\displaystyle \gamma }$ such that

${\displaystyle |\alpha \,B\,f\rangle =|\,B\,f\rangle \,\gamma \,\alpha \,\gamma ^{-1}\,{\overset {\underset {\mathrm {def} }{}}{=}}\,|B\,\gamma \,\alpha \,\gamma ^{-1}\,f\rangle }$

(14 )

If ${\displaystyle |a\rangle }$ is an eigenvector of operator ${\displaystyle A}$ with quaternionic eigenvalue ${\displaystyle \alpha }$

${\displaystyle A|a\rangle =|a\rangle \alpha }$

(15 )

then ${\displaystyle |\beta a\rangle }$ is an eigenvector of ${\displaystyle A}$ with quaternionic eigenvalue ${\displaystyle \beta ^{-1}\alpha \beta }$.

${\displaystyle A|\beta a\rangle =A|a\rangle \beta =|a\rangle \alpha \beta =|\beta a\rangle \beta ^{-1}\alpha \beta }$

(16 )

${\displaystyle A^{\dagger }}$ is the adjoint of the normal operator ${\displaystyle A}$.

${\displaystyle \langle f|Ag\rangle =\langle fA^{\dagger }|g\rangle =\langle g|A^{\dagger }f\rangle ^{*}}$

(17 )

${\displaystyle A^{\dagger \dagger }=A}$

(18 )

${\displaystyle (A+B)^{\dagger }=A^{\dagger }+B^{\dagger }}$

(19 )

${\displaystyle (A\,B)^{\dagger }=B^{\dagger }A^{\dagger }}$

(20 )

If ${\displaystyle A=A^{\dagger }}$ , then ${\displaystyle A}$ is a self adjoint operator.

| is a nil operator.

A linear transformation ${\displaystyle L}$ of Hilbert space ${\displaystyle {\mathfrak {H}}}$ changes the value of the inner product with the transformed vector.

${\displaystyle s_{L}=\langle f|L\,g\rangle }$

(21 )

The effect of the transpose transformation ${\displaystyle L^{\dagger }}$ is then given by

${\displaystyle \langle f|L\,g\rangle =\langle L^{\dagger }f|g\rangle }$

(22 )

A linear operator is normal if ${\displaystyle L^{\dagger }L}$ exists and ${\displaystyle L^{\dagger }L=L\ L^{\dagger }}$.

For a normal transformation ${\displaystyle N}$ holds

${\displaystyle \langle Nf|N\,g\rangle =\langle N^{\dagger }Nf|g\rangle =\langle NN^{\dagger }f|g\rangle =\langle f|N^{\dagger }Ng\rangle =\langle f|NN^{\dagger }g\rangle }$

(23 )

Thus

${\displaystyle N=N_{r}+{\vec {N}}}$

(24 )

${\displaystyle N^{\dagger }=N_{r}-{\vec {N}}}$

(25 )

${\displaystyle N_{r}={\frac {N+N^{\dagger }}{2}}}$

(26 )

${\displaystyle {\vec {N}}={\frac {N-N^{\dagger }}{2}}}$

(27 )

${\displaystyle NN^{\dagger }=N^{\dagger }N=N_{r}N_{r}+\langle {\vec {N}},{\vec {N}}\rangle =|N|^{2}}$

(28 )

${\displaystyle N_{r}}$ is the Hermitian part of ${\displaystyle N}$.

${\displaystyle {\vec {N}}}$ is the anti-Hermitian part of ${\displaystyle N}$.

For two normal operators ${\displaystyle A}$ and ${\displaystyle B}$ holds

${\displaystyle A\,B=A_{r}B_{r}-\langle {\vec {A}},{\vec {B}}\rangle +A_{r}{\vec {B}}+{\vec {A}}B_{r}\pm {\vec {A}}\times {\vec {B}}}$

(29 )

For a unitary transformation ${\displaystyle U}$ holds

${\displaystyle \langle Uf|U\,g\rangle =\langle f|g\rangle }$

(30 )

##### Closure

The closure of ${\displaystyle {\mathfrak {H}}}$ means that converging rows of vectors converge to a vector of ${\displaystyle {\mathfrak {H}}}$.

#### Operator construction

${\displaystyle |f\rangle \langle g|}$ is a linear operator. ${\displaystyle |g\rangle \langle f|}$ is its adjoint operator.

For the orthonormal base ${\displaystyle \{|q_{i}\rangle \}}$

holds

${\displaystyle \langle q_{n}|q_{m}\rangle =\delta (n,m)}$

(31 )

${\displaystyle \langle g|{\color {Red}F}\,h\rangle =\sum _{i=1}^{N}\{\langle g{\color {Red}|q_{i}\rangle F(q_{i})\langle q_{i}|}h\rangle \}}$

(32 )

This definition enables the shorthand

${\displaystyle F\ {\overset {\underset {\mathrm {def} }{}}{=}}\ |q_{i}\rangle F(q_{i})\langle q_{i}|}$

(33 )

It is evident that

${\displaystyle F^{\dagger }\ {\overset {\underset {\mathrm {def} }{}}{=}}\ |q_{i}\rangle F^{*}(q_{i})\langle q_{i}|}$

(34 )

Reference operator ${\displaystyle {\mathfrak {R}}^{x}}$ holds

${\displaystyle {\mathfrak {R}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ |q_{i}\rangle q_{i}\langle q_{i}|}$

(35 )

If ${\displaystyle \{q_{i}^{x}\}}$ consists of all rational values of a version of the quaternionic number system (marked by superscript ${\displaystyle ^{x}}$) then ${\displaystyle {\mathfrak {R}}^{x}}$ represents the parameter space of (discrete) function ${\displaystyle F(q_{i}^{x})}$.

${\displaystyle {\mathfrak {R}}^{x}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ |q_{i}^{x}\rangle q_{i}^{x}\langle q_{i}^{x}|}$

(36 )

Omitting the superscript ${\displaystyle ^{x}}$ means that we take numbers from the version of the number system that serves the specification of the inner product.

We call the combination of equation 32 and equation 33 the reverse bra-ket method.

## Non-separable Hilbert space

Every infinite dimensional separable Hilbert space ${\displaystyle {\mathfrak {H}}}$ owns a unique non-separable companion Hilbert space ${\displaystyle {\mathcal {H}}}$. This is achieved by the closure of the eigenspaces of all reference operators. In this procedure, in many occasions, the notion of the dimension of subspaces looses its sense.

Gelfand triple and Rigged Hilbert space are other names for the general non-separable Hilbert spaces.

In the non-separable Hilbert space, for operators with continuum eigenspaces, the reverse bra-ket method turns from a summation into an integration

${\displaystyle \langle g|{\color {Red}F}\,h\rangle {\overset {\underset {\mathrm {def} }{}}{=}}\iiiint \ \{\langle g{\color {Red}|q\rangle F(q)\langle q|}h\rangle \}dVd\tau }$

(37 )

Here, the subscripts of the countable base vanished.

The corresponding shorthand for operator ${\displaystyle F}$ is

${\displaystyle F\ {\overset {\underset {\mathrm {def} }{}}{=}}\ |q\rangle F(q)\langle q|}$

(38 )

For eigenvectors ${\displaystyle |q\rangle }$ the function ${\displaystyle F(q)}$ defines as

${\displaystyle F(q)=\langle q|F\,q\rangle =\int \limits _{q'}\langle q|q'\rangle F(q')\langle q'|q\rangle dq'}$

(38a )

The reference operator ${\displaystyle {\mathcal {R}}}$ that provides the background parameter space as its eigenspace follows from

${\displaystyle

(39 )

with shorthand

${\displaystyle {\mathcal {R}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ |q\rangle q\langle q|}$

(40 )

### Floating Platforms

The non-separable Hilbert space can use the same versions of the quaternionic number system to define a series of parameter spaces that are eigenspaces of corresponding reference operators as its separable companion did. One of these parameter spaces is special because it applies the version of the quaternionic number system that serves the specification of the inner product of the companion Hilbert spaces. The corresponding eigenspace delivers the background parameter space of the Hilbert spaces. All other reference operators deliver eigenspaces that float with respect to the background parameter space and that possess a private ordering symmetry. A Cartesian coordinate system in combination with a polar coordinate system specifies the ordering symmetry. The difference between the ordering symmetry of the parameter space with the ordering symmetry of the background parameter space determines the symmetry flavor of the considered parameter space. The symmetry flavor delivers a symmetry related charge that locates at the geometric center of the floating parameter space. This symmetry related charge interacts with a symmetry related field.

The floating parameter spaces and their symmetry-related charges belong to the furniture of the separable Hilbert space. The embedding of the separable Hilbert space into the non-separable Hilbert space introduces an interaction with the symmetry related field.

### Scanning subspace

The eigenvectors of the reference operator that defines the background parameter space can be split with respect to a selected progression value that specifies the real parts of the eigenvalues. The eigenvectors that correspond to the progression value span a scanning subspace that represents a static status quo. Lower values of the real part of the eigenvalue define the historical part of the Hilbert space. Higher values define the future part of the Hilbert space. This scanning subspace turns a base model that consists of a separable Hilbert space and its companion non-separable Hilbert space into a dynamic model.

The separable Hilbert space ${\displaystyle {\mathfrak {H}}}$ and its non-separable companion Hilbert space ${\displaystyle {\mathcal {H}}}$ combine into a base model in which ${\displaystyle {\mathcal {H}}}$ embeds ${\displaystyle {\mathfrak {H}}}$ via an embedding process that takes place in the scanning subspace ᙎ.

Progression ${\displaystyle \tau }$ steps with in ${\displaystyle {\mathfrak {H}}}$ and flows in ${\displaystyle {\mathcal {H}}}$.

This base model must extend with extra mechanisms that specify the dynamic locations of objects that hop around on the floating platforms. These extra mechanisms form the subject of the full Hilbert Book Model ᙢ.

### Merging technologies

The combination of the two Hilbert spaces ${\displaystyle {\mathfrak {H}}}$ and ${\displaystyle {\mathcal {H}}}$ together with the reverse bra-ket method for the definition of operators via reference operators and quaternionic functions, creates a powerful base model that merges quaternionic Hilbert space operator technology with quaternionic function theory and indirectly with quaternionic differential and integral calculus.

The combination enables to consider that the non-separable Hilbert space ${\displaystyle {\mathcal {H}}}$ embeds the separable Hilbert space ${\displaystyle {\mathfrak {H}}}$ in an ongoing process that acts as a function of the progress of a scanning subspace that defines the instant of the embedding.

As long as a differentiation or integration results in a sufficiently continuous function, then this function can help to define a new operator. This offers the opportunity to represent the solutions of differential and integral equations inside the base model .

Also, the results of stochastic processes can be stored in the separable Hilbert space ${\displaystyle {\mathfrak {H}}}$ and if the result forms a coherent and dense location swarm, then the corresponding location density distribution can store embedded in a continuum in the non-separable Hilbert space ${\displaystyle {\mathcal {H}}}$. This is applied in the Hilbert Book Model .

### Sets, coherent swarms, distributions and embedding continuums

The operators in the separable Hilbert space offer eigenspaces that can be

• Sets of spurious eigenvalues
• Stochastic coherent swarms of eigenvalues
• Stochastic coherent swarms of eigenvalues can be considered to be generated by a stochastic process that owns a characteristic function
• Stochastic coherent swarms of eigenvalues may be ordered with respect to their timestamp and become well ordered coherent swarms
• Stochastic well ordered coherent swarms of eigenvalues are ordered with respect to their timestamp
• Stochastic well ordered coherent swarms of eigenvalues may be spatially ordered and then become ordered distributions of eigenvalues
• Ordered distributions of eigenvalues
• Ordered distributions posses a location density distribution

If a continuum embeds an ordered distribution, then the defining function of the operator that has the continuum as eigenspace is the convolution of the location density distribution of the ordered distribution with the Green'sfunction of the continuum.

For example, an ordered distribution that is defined by an isotropic Gaussian function results in an embedding continuum that is definened by an ${\displaystyle {\frac {{\mathsf {ERF}}(r)}{r}}}$ function.