Hilbert space

A Hilbert space (also Hilbert space ), named after the German mathematician David Hilbert, is a concept from the mathematical branch of functional analysis. A Hilbert space is a vector space over the real or complex numbers, equipped with a scalar product - and thus angle and length terms - which is complete with respect to the scalar product induced by the norm ( length of the term). A Hilbert space is a Banach space whose norm is induced by an inner product. If you drop the condition of completeness, it is called a pre-Hilbert space.

The structure of a Hilbert space is uniquely determined by its Hilbert space dimension. This can be any cardinal number. If the dimension is finite, then it is a Euclidean space. In many areas, such as in the mathematical description of quantum mechanics, is "the" Hilbert space with a countable dimension, that is, with the smallest possible infinite dimension, of particular importance. An element of a Hilbert space can ( called Cartesian coordinates in the finite ) as a family of one of the dimension corresponding number of real or complex values ​​are interpreted. Similar to vector spaces, whose elements are always only finitely many coordinates of a Hamel basis of non-zero, each element of a Hilbert space only countably many coordinates an orthonormal basis equal to zero and the coordinate family is quadratsummabel.

Hilbert spaces contribute by their scalar topological structure characterized here as opposed to general vector spaces limit processes possible. Hilbert spaces are closed under countable sums of orthogonal elements with a quadratsummablen series of standards or of parallel elements with a absolutsummablen series of standards.

Definition

A Hilbert space is a real or complex vector space with a scalar product, which is complete with respect to the induced by the scalar product norm in which every Cauchy sequence converges. A Hilbert space is thus a complete pre-Hilbert space.

Hereinafter, the dot product is linear and in the second semi- linearly in the first argument, i.e., a complex vector space and are vectors and a complex number, it is

In which case the scalar product is semi- linear, convention and is often handled differently around.

Importance

Hilbert spaces play in the functional analysis, especially in the solution theory of partial differential equations, and thus also in physics a major role. As an example we mention the quantum mechanics where the states of a quantum- theoretical system form a Hilbert space. From the perspective of functional analysis, the Hilbert spaces form a class of spaces with very specific and simple structure.

Examples of Hilbert spaces

  • The coordinate space with the standard scalar real.
  • The coordinate space with the standard scalar complex.
  • The die space of real or complex matrices with the Frobenius inner product.
  • Subsequently, the space of all sequences with the property that the sum of the squares of all the sequence elements is finite. This is the original Hilbert space, by which David Hilbert examined the properties of such spaces. Further, this example is important because all separable infinite-dimensional Hilbert spaces are isometrically isomorphic to.
  • The space of square integrable functions with the inner product. For a complete definition, which examined more closely in particular the completeness, can be found in the article on Lp- spaces.
  • The space of the quasi- periodic functions, which is defined as follows: to consider one of the functions. By the scalar product is the space ( spanned by the subspace of the space of all functions functions ) to a pre-Hilbert space. The completion of this space is thus a Hilbert space. In contrast to the above examples, this area is not separable.
  • The Sobolev space for all and the corresponding subspaces. These form a basis of the solution theory of partial differential equations.
  • The space of the Hilbert-Schmidt operators.
  • For are the Hardy space and the real Hardy space Hilbert spaces.

Orthogonality and orthogonal systems

Two elements of the Hilbert space are called orthogonal if their dot product yields 0. A family of pairwise orthogonal vectors is called orthogonal. Among the Orthogonalsystemen the Orthogonalbasen play a special role: they are orthogonal systems, which can not be enlarged by adding another vector, so are maximal with respect to inclusion. Equivalent to this is that the linear span is dense in the Hilbert space. Except in the case of finite-dimensional spaces form Orthogonalbasen no basis in the usual sense of linear algebra ( Hamel basis). Are these basis vectors beyond normalized so that the dot product of a vector with itself yields 1, then one speaks of a orthonormal system or an orthonormal basis. Thus, the vectors form an orthonormal system if and only if for all. It is the Kronecker delta.

Using the lemma of anger can be shown that every Hilbert space has an orthonormal basis ( it may even be complements any orthonormal system to an orthonormal basis).

Subspaces

A sub- Hilbert space or part of the Hilbert space of a Hilbert space is a subset restricted to the scalar multiplication, addition and scalar product in turn forms in this subset of a Hilbert space. Specifically, this means that the subset contains the zero and closed under scalar multiplication and addition, that is a subspace, and with respect to the scalar product is still incomplete. This is equivalent to saying that the subset is closed in the topological sense. Therefore, one designated sub ​​Hilbert spaces as closed subspaces and closed subspaces and referred contrast, any subspaces just as subspaces and subspaces. Such a man is only a pre-Hilbert space in general. Each pre-Hilbert space is contained in a Hilbert space as a dense subspace, namely in its completion. It is also possible to form a space with respect to the ratio of a sub Hilbert space, which in turn is a Hilbert space.

All this applies essentially analogous for arbitrary Banach spaces, where the subspaces then not necessarily Prähilberträume, but rather are normed spaces. A special feature, however, is the validity of the projection set: For each sub- Hilbert space and any element of the Hilbert space there is an element of the Hilbert space with minimal distance. This is true for Banach spaces, however, already in the finite generally not. This permits identification of the canonical space ratio with respect to a sub Hilbert space with a lower Hilbert space, the orthogonal complement, and the concept of orthogonal. The orthogonal complement of a sub- Hilbert space is a complementary sub- Hilbert space, for Banach spaces, however, there is a Unterbanachraum generally not complementary Unterbanachraum.

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 and in the second component in the first linear conjugated. One can therefore define a complex Hilbert space as follows another Hilbert space. When quantity is also the addition to be adopted by. The scalar multiplication and the dot product for are explained as follows:

It checks to see that with these definitions is again a Hilbert space, it is called the conjugate Hilbert space. The conjugated to Hilbert space is apparently back.

Operators between Hilbert spaces

Rich object of study in functional analysis are also some structure-preserving maps between Hilbert spaces. Mainly one considers it figures that get the vector space structure, ie linear maps, hereinafter referred to as linear operators.

An important class of linear operators between Hilbert spaces is that of continuous operators, in addition the topological structure, and thus about limits obtained. Other important classes of linear operators arise from the fact that one of them requires certain boundedness properties. The continuity is a general tendency for normed spaces, equivalent to the boundedness of the operator. A stronger restriction is the compactness. The shadow classes are true subclasses of the class of compact operators. On the respective classes of operators different standards and operator topologies are defined.

Unitary operators provide a natural Isomorphismenbegriff for Hilbert spaces, they are just the isomorphisms in the category of Hilbert spaces with the linear contractions as morphisms. Specifically: the linear surjective isometries. You get all the lengths and angles. From the set of Fréchet - Riesz also follows that the adjoint operator can be understood to be a linear operator from to as a linear operator from to. This allows an operator that commutes with its adjoint, such operators form the class of normal operators. For operators in a Hilbert space there is the possibility that the adjoint operator in turn is the operator itself, then one speaks of a self-adjoint operator.

Many of the above-mentioned classes of operators form restricted to operators on a single Hilbert space operator algebras. With the Adjungierung as involution, under which all classes listed above have been completed, and a matching standard involutive Banach algebras arise even. The continuous linear operators on a Hilbert space with the Adjungierung and the operator norm form a C * - algebra.

Classification

Using orthonormal bases can be Hilbert spaces completely classify. Every Hilbert space has an orthonormal basis, and any two orthonormal bases of a Hilbert space are equally powerful. The cardinality of any orthonormal basis is thus a well-defined property of a Hilbert space, which Hilbert space dimension or short dimension is called. The two Hilbert spaces with the same dimension are isomorphic: This gives an isomorphism by clearly continues 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. Indeed, there is at any cardinal number a Hilbert space with this dimension, constructible about as a room (with a lot of dimension cardinality is, about the cardinal number itself):

Where or and the convergence of the sum is to be read so that only countably many summands are equal ( cf. unconditional convergence). This space is equipped with the scalar product

Which is well defined. The vectors then form an orthonormal basis of the space. The isomorphism of each Hilbert space with such a room for matching is known as a set of Fischer- Riesz.

Dual space

The topological dual space of continuous linear functionals on a Hilbert space is a Banach space again, as with any Banach space itself. A special feature of Hilbert spaces is the set of Fréchet - Riesz: Every real Hilbert space is isomorphic by means of an isometric Vektorraumisomorphismus to its dual space. The norm on the dual space is therefore also be induced by a scalar product, it is thus also a Hilbert space. In the case of a complex Hilbert space, the set is valid analogy, but that figure is only semi -linear, that is, a antiunitärer operator. In both cases, the Hilbert space is isomorphic to its dual space ( a antiunitärer operator can in fact into a unitary operator and a antiunitary operator disassemble ), and thus a fortiori to his Bidualraum, every Hilbert space is thus reflexive.

Fourier coefficient

An orthonormal basis is a powerful tool in the study of Hilbert spaces over, respectively, and their elements. Particularly an orthonormal basis provides an easy way of determining the representation of a vector by the elements of the orthonormal basis. Be an orthonormal basis and a vector of the Hilbert space. Since a Hilbert space basis of the space forms, there are coefficients and so

Is. These coefficients are determined by the orthonormal basis taking advantage of the special properties as

As the scalar product of different basis vectors 0 and of the same base vectors is 1. The - th base coefficient of the representation of a vector in an orthonormal basis can therefore be determined by scalar product. These coefficients are also referred to as Fourier coefficients, since they represent a generalization of the concept of the Fourier analysis.

Trivia

At several German universities there as " Hilbert space " designated premises, for example at the Universities of Dortmund, Frankfurt, Konstanz, Mainz and at the Georg -August- University Göttingen, at the David Hilbert for many years taught and researched; there bears the foyer of the Mathematical Institute, where a bust of the mathematician is placed that name.

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