Hilbert dream

A Hilbert space (also Hilbert space, Hilbertscher space ), named after the German mathematician David Hilbert , is a term from the mathematical branch of functional analysis . A Hilbert space is a vector space over the field of real or complex numbers , provided with a scalar product - and therefore the angle and length terms - which completely with respect to the induced by the inner product standard (of the length term). A Hilbert space is a Banach space whose norm is induced by a scalar product. If the condition of completeness is dropped, one speaks of a prehilbert dream .

The structure of a Hilbert space is clearly defined by its Hilbert space dimension . This can be any cardinal number . If the dimension is finite and if the real numbers are considered as the body, then it is a question of Euclidean space . In many areas, for example in the mathematical description of quantum mechanics , “the” Hilbert space has a countable dimension, i. H. with the smallest possible infinite dimension, of particular importance. An element of a Hilbert space can be understood as a family of a number of real or complex values ​​corresponding to the dimension ( called Cartesian coordinates in finite dimensions ). Analogous to vector spaces, the elements of which are only non-zero in a finite number of coordinates of a Hamel basis , each element of a Hilbert space is only non-zero in a countable number of coordinates of an orthonormal basis and the coordinate family is square-sumable .

Hilbert spaces have a topological structure through their scalar product . As a result, in contrast to general vector spaces, limit value processes are possible here. Hilbert spaces are closed under countable sums of orthogonal elements with a square summable sequence of norms or of parallel elements with an absolute summable sequence of norms.

definition

A Hilbert space is a real or complex vector space with a scalar product , which is completely with respect to the norm induced by the scalar product , in which every Cauchy sequence converges . A Hilbert dream is therefore a complete Prehilbert dream. ${\displaystyle H}$ ${\displaystyle \langle \cdot ,\cdot \rangle }$

In the following, let the scalar product be linear in the second and semilinear in the first argument, i.e. H. is a complex vector space and if vectors and a scalar (complex number) are, so is ${\displaystyle H}$ ${\displaystyle u,v\in H}$ ${\displaystyle \lambda \in \mathbb {C} }$

${\displaystyle \langle u,\lambda v\rangle =\lambda \langle u,v\rangle }$and . ${\displaystyle \langle \lambda u,v\rangle ={\bar {\lambda }}\langle u,v\rangle }$

In which argument the scalar product is semilinear is convention and is often handled the other way around.

meaning

Hilbert spaces play a major role in functional analysis , especially in the solution theory of partial differential equations , and thus also in physics . One example is quantum mechanics , where pure states of a quantum mechanical system can be described by a vector in Hilbert space. From the point of view of functional analysis, the Hilbert spaces form a class of spaces with a particularly special and simple structure.

Examples of Hilbert spaces

• The coordinate space with the standard real scalar product . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \langle u,v\rangle =u_{1}v_{1}+\dotsb +u_{n}v_{n}}$

• The coordinate space with the complex standard scalar product . ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle \langle u,v\rangle ={\bar {u}}_{1}v_{1}+\dotsb +{\bar {u}}_{n}v_{n}}$

• The matrix space of the real or complex matrices with the Frobenius scalar product . ${\displaystyle {\mathbb {K} }^{m\times n}}$

• The sequence space of all sequences with the property that the sum of the squares of all sequence members is finite. This is the original Hilbert space, on the basis of which David Hilbert examined the properties of such spaces. Further, this example is important because all separable infinite-dimensional Hilbert space isometrically isomorphic to have. ${\displaystyle \ell ^{2}}$ ${\displaystyle \ell ^{2}}$

• The space of the square integrable functions with the scalar product . A complete definition, which particularly highlights completeness, can be found in the article on L ^p spaces . ${\displaystyle L^{2}}$ ${\displaystyle \textstyle \langle f,g\rangle _{L^{2}}=\int {\overline {f(x)}}\,g(x)\,{\rm {d}}x}$

• The space of almost periodic functions, which is defined as follows: To consider the functions with . Through the scalar product , the space (the subspace of the space of all functions spanned by the functions) becomes a prehilbert space. The completion of this space is therefore a Hilbert space. In contrast to the examples above, this room is not separable. ${\displaystyle \mathrm {AP} ^{2}}$ ${\displaystyle \lambda \in \mathbb {R} }$ ${\displaystyle f_{\lambda }\colon \mathbb {R} \to \mathbb {C} }$ ${\displaystyle f_{\lambda }\left(t\right)=e^{i\lambda t}}$ ${\displaystyle \textstyle \langle f,g\rangle =\lim _{T\to +\infty }{\tfrac {1}{4T}}\int _{-T}^{T}{\overline {f(t)}}\,g(t)\,{\rm {d}}t}$ ${\displaystyle \operatorname {lin} \left\{f_{\lambda }\colon \lambda \in \mathbb {R} \right\}}$ ${\displaystyle f_{\lambda }}$ ${\displaystyle \mathrm {AP} ^{2}}$

• The Sobolev room for everyone and the corresponding sub-rooms. These form a basis of the solution theory of partial differential equations . ${\displaystyle H^{p}}$ ${\displaystyle p\geq 0}$

• The space of the Hilbert-Schmidt operators . ${\displaystyle HS}$

• For the Hardy space and the real Hardy space are Hilbert spaces. ${\displaystyle p=2}$ ${\displaystyle H^{2}(\mathbb {D} )}$ ${\displaystyle {\mathcal {H}}^{2}(\mathbb {R} ^{n})}$

Orthogonality and Orthogonal Systems

Two elements of the Hilbert space are called orthogonal to each other if their scalar product is 0. A family of pairwise orthogonal vectors is called an orthogonal system. Among the orthogonal systems, the orthogonal bases play a special role: these are orthogonal systems that can no longer be enlarged by adding a further vector, i.e. are maximal in terms of inclusion. Equivalent to this is that the linear hull in Hilbert space is dense. Except in the case of finite-dimensional spaces, orthogonal bases do not form a basis in the usual sense of linear algebra ( Hamel basis ). If these basis vectors are also normalized in such a way that the scalar product of a vector with itself results in 1, one speaks of an orthonormal system or an orthonormal basis. The vectors form an orthonormal system if and only if for all . This is the Kronecker Delta . ${\displaystyle v_{i}}$ ${\displaystyle \langle v_{i},v_{j}\rangle =\delta _{ij}}$ ${\displaystyle i,j}$ ${\displaystyle \delta _{ij}}$

Using Zorn's lemma, it can be shown that every Hilbert space has an orthonormal basis (every orthonormal system can even be supplemented to an orthonormal basis).

Subspaces

A Unterhilbert space or partial Hilbert space of a Hilbert space is a subset which in turn forms a Hilbert space with the scalar multiplication , addition and scalar product restricted to this subset. In concrete terms, this means that the subset contains the zero and is closed under scalar multiplication and addition, that is, is a sub-vector space , and is still complete with respect to the scalar product. This is equivalent to that subset in the topological sense finished is. Therefore, sub-spaces are also referred to as closed sub-spaces or closed sub-spaces and, in contrast, any sub-vector spaces are simply referred to as sub-spaces or sub-spaces . Such is generally just a prehistoric dream. Every Prähilbert space is contained in a Hilbert space as a dense sub-vector space, namely in its completion . It is also possible to form a quotient space with respect to a Unterhilbert space, which in turn is a Hilbert space.

All of this essentially applies analogously to any Banach spaces , whereby their sub-vector spaces are then not necessarily Prähilbert spaces, but normalized spaces . A special feature, however, is the validity of the projection theorem : for every Unterhilbert space and every element of the Hilbert space there is an element of the Unterhilbert space with a minimal distance. On the other hand, this does not apply to Banach spaces in the finite-dimensional space in general. This allows a canonical identification of the quotient space with respect to a Unterhilbert space with a Unterhilbert space, the orthogonal complement , and the concept of orthogonal projection . The orthogonal complement of an Unterhilbert space is a complementary Unterhilbert space, for Banach spaces, on the other hand, there is generally no complementary Unterhilbert space to a Unterhilbert space.

Conjugated Hilbert space

In the case of a complex Hilbert space there is a certain asymmetry between the two components of the scalar product; the scalar product is linear in the second component and conjugate linear in the first. One can therefore define another Hilbert space for a complex Hilbert space as follows . As a quantity , the addition to is also taken over by. The scalar multiplication and the scalar product for are explained as follows: ${\displaystyle H}$ ${\displaystyle {\overline {H}}}$ ${\displaystyle {\overline {H}}=H}$ ${\displaystyle {\overline {H}}}$ ${\displaystyle H}$ ${\displaystyle {\overline {H}}}$

scalar multiplication: ${\displaystyle \lambda \cdot _{\overline {H}}u:={\overline {\lambda }}u}$

Scalar product: . ${\displaystyle \langle u,v\rangle _{\overline {H}}:={\overline {\langle u,v\rangle }}=\langle v,u\rangle }$

You check that with these definitions there is again a Hilbert space; it is called the conjugated Hilbert space . The Hilbert space to be conjugated is obviously again . ${\displaystyle {\overline {H}}}$ ${\displaystyle {\overline {H}}}$ ${\displaystyle H}$

Operators between Hilbert spaces

→ Main article : Linear operator

Extensive objects of investigation in functional analysis are also certain structure-preserving images between Hilbert spaces. Here, one mainly considers mappings which contain the vector space structure, that is to say linear mappings , hereinafter referred to as linear operators .

An important class of linear operators between Hilbert spaces is that of the continuous operators , which also receive the topological structure and thus, for example, limit values. Further important classes of linear operators result from the fact that certain boundedness properties are required of them. As is generally the case with normalized spaces, continuity is equivalent to the boundedness of the operator. A major limitation is that of compactness . The shadow classes are real sub-classes of the class of compact operators. Different norms and operator topologies are defined on the respective classes of operators .

Unitary operators provide a natural concept of isomorphism for Hilbert spaces; they are precisely the isomorphisms in the category of Hilbert spaces with the linear mappings that receive the scalar product as morphisms . Specifically: the linear, surjective isometrics. You get all lengths and angles. It also follows from the Fréchet-Riesz theorem that the adjoint operator to a linear operator from to can be understood as a linear operator from to . This allows an operator to commute with its adjoint operator , such operators form the class of normal operators . In the case of operators within a Hilbert space, the possibility arises that the adjoint operator is in turn the operator itself; one then speaks of a self-adjoint operator . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Y}$ ${\displaystyle X}$

Many of the classes of operators listed above form restricted to operators on a single Hilbert space operator algebras . With the adjoint as involution , under which all classes listed above are closed, and a suitable norm, even involutive Banach algebras result . The continuous linear operators on a Hilbert space with adjoint and the operator norm form a C * algebra .

classification

→ Main article : Theorem from Fischer-Riesz

The Hilbert spaces can be fully classified using orthonormal bases. Every Hilbert space has an orthonormal basis and two orthonormal bases of a Hilbert space are equally powerful . The cardinality of every orthonormal basis is therefore a well-defined property of a Hilbert space, which is called the Hilbert space dimension or dimension for short . Every two Hilbert spaces with the same dimension are isomorphic : An isomorphism is obtained by unambiguously continuing a bijection between an orthonormal basis of one and an orthonormal basis of the other to a continuous linear operator between the spaces. Every continuous linear operator between two Hilbert spaces is uniquely determined by its values ​​on an orthonormal basis of the space on which it is defined. In fact, for every cardinal number there is a Hilbert space with this dimension, which can be constructed as a space (where a set with the dimension as cardinality, e.g. the cardinal number itself): ${\displaystyle \ell ^{2}(I)}$ ${\displaystyle I}$

${\displaystyle \ell ^{2}(I):=\left\{u\colon I\to K\mid \sum _{i\in I}\left|u(i)\right|^{2}<\infty \right\}}$,

where or and the convergence of the sum is to be read in such a way that only a countable number of summands are unequal (cf. unconditional convergence ). This space is provided with the scalar product ${\displaystyle K=\mathbb {R} }$ ${\displaystyle K=\mathbb {C} }$ ${\displaystyle 0}$

${\displaystyle \langle u,v\rangle :=\sum _{i\in I}{\overline {u(i)}}v(i)}$,

which is well defined. The vectors with then form an orthonormal basis of the space . The isomorphism of any Hilbert space with such a space for something suitable is known as the Fischer-Riesz theorem . ${\displaystyle u_{i}}$ ${\displaystyle u_{i}(j)=\delta _{ij}}$ ${\displaystyle \ell ^{2}(I)}$ ${\displaystyle \ell ^{2}(I)}$ ${\displaystyle I}$

Dual space

The topological dual space of the continuous, linear functionals on a Hilbert space is itself a Banach space as with every Banach space. A special feature of Hilbert spaces is the Fréchet-Riesz theorem : Every real Hilbert space is isomorphic to its dual space by means of the isometric vector space isomorphism. The norm on the dual space is therefore also induced by a scalar product, so it is also a Hilbert space. In the case of a complex Hilbert space, the theorem applies analogously, but the mapping is only semilinear , that is, an antiunitary operator . In both cases the Hilbert space is isomorphic to its dual space (an anti-unit operator can be divided into a unitary operator and an anti-unit operator ), and thus even more so to its dual space , so every Hilbert space is reflexive . ${\displaystyle H^{\prime }}$ ${\displaystyle H}$ ${\displaystyle H}$ ${\displaystyle H\rightarrow H^{\prime },\,v\mapsto \langle v,\cdot \rangle }$ ${\displaystyle H\to H^{\prime }}$ ${\displaystyle H\to H^{\prime }}$ ${\displaystyle H^{\prime }\to H^{\prime }}$

Fourier coefficient

An orthonormal basis is a powerful tool for the investigation of Hilbert spaces over or and its elements. In particular, an orthonormal basis offers a simple possibility of determining the representation of a vector by the elements of the orthonormal basis. Let be an orthonormal basis and a vector from the Hilbert space. Since a Hilbert space basis forms the space, there are coefficients or , such that ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle B=(b_{1},b_{2},\dots )}$ ${\displaystyle v}$ ${\displaystyle B}$ ${\displaystyle \alpha _{k}\in \mathbb {R} }$ ${\displaystyle \mathbb {C} }$

${\displaystyle v=\sum _{k}\alpha _{k}b_{k}}$

is. These coefficients are determined using the special properties of the orthonormal basis as

${\displaystyle \langle b_{n},v\rangle =\left\langle b_{n},\sum _{k}\alpha _{k}b_{k}\right\rangle =\sum _{k}\alpha _{k}\langle b_{n},b_{k}\rangle =\alpha _{n}}$,

since the scalar product of different basis vectors is 0 and of the same basis vectors 1. The -th basic coefficient of the representation of a vector in an orthonormal basis can thus be determined by forming scalar products. These coefficients are also called Fourier coefficients because they represent a generalization of the concept of Fourier analysis . ${\displaystyle n}$

RKHS

If one associates a Hilbert space with a core that reproduces every function within the space, one speaks of a Reproducing Kernel Hilbert Space (RKHS, German: Hilbertraum mit reproduzierendem Kern). This approach was first formulated in 1907 by the mathematician Stanisław Zaremba and began to play an important role in functional analysis half a century later . Today Hilbert spaces with a reproducing core are a common tool in statistical learning theory, especially in machine learning .

Hilbert spaces in quantum mechanics

The axioms of quantum mechanics state that the set of possible states of a quantum mechanical system has the structure of a Hilbert space. In particular, this means that quantum mechanical states have a linear structure, i.e. that a linear combination of states again results in a physically possible state. In addition, a scalar product between two states and is defined, the absolute square of which, according to Born's probability interpretation, indicates how likely it is to find a system that is in the state during a measurement in the state . (The notation corresponds to the Dirac notation .) When physics speaks of the Hilbert space, it means the state space of the given quantum mechanical system. ${\displaystyle \langle \psi |\phi \rangle }$ ${\displaystyle |\psi \rangle }$ ${\displaystyle |\phi \rangle }$ ${\displaystyle |\phi \rangle }$ ${\displaystyle |\psi \rangle }$

examples are

• the possible wave functions of a free particle are the Hilbert space of all square-integrable functions with the usual -Scalar product . ${\displaystyle L^{2}}$ ${\displaystyle \psi \colon \mathbb {R} ^{3}\rightarrow \mathbb {C} }$ ${\displaystyle L^{2}}$ ${\displaystyle \textstyle \langle \psi \,|\,\phi \rangle =\int _{\mathbb {R} ^{3}}\psi ^{*}({\vec {x}})\,\phi ({\vec {x}})\,{\rm {d}}{\vec {x}}}$
• the possible spin states of an electron span the Hilbert space with the Euclidean scalar product. ${\displaystyle \mathbb {C} ^{2}}$

Trivia

At several universities in German-speaking countries there are rooms known as “Hilbertraum”.

literature

• Dirk Werner : Functional Analysis. 5th enlarged edition. Springer, Berlin et al. 2005, ISBN 3-540-43586-7 , Chapters V, VI and VII.
• Richard V. Kadison , John R. Ringrose : Fundamentals of the Theory of Operator Algebras. Volume 1: Elementary Theory. Academic Press, New York NY 1983, ISBN 0-12-393301-3 ( Pure and Applied Mathematics 100, 1), Chapter 2: Basics of Hilbert Space and Linear Operators .

See also

• Bessel's inequality
• Cauchy-Schwarz inequality
• Hilbert space base
• Hilbert space tensor product
• Parallelogram equation
• Parseval's equation
• Peetre's inequality

Individual evidence

1. ^ Hilbert room of the Mathematics Student Council at the University of Konstanz. Retrieved October 8, 2019 .  
2. ^ Friends of Mathematics at Johannes Gutenberg University Mainz, event Mathematics and School. Retrieved October 8, 2019 .  
3. ^ Hilbert room of the physics student council at the Technical University of Dortmund. Retrieved May 24, 2020 .