Banach space

A Banach space (also Banach space ) is a complete normed vector space in mathematics. Banach spaces are central objects of study in functional analysis. In particular, many infinite-dimensional function spaces are Banach spaces. They are named after the mathematician Stefan Banach, who introduced her 1920-1922 together with Hans Hahn and Eduard Helly.

Definition

A Banach space is a complete normed space

That is, a vector space over the field of real or complex numbers with a norm in which every Cauchy sequence converges from elements of the induced from the standard metric.

Notes

Wherein the integrity metric spaces is a property of the metric of the non- topological space itself is passed to a metric equivalent (i.e., a metric that produces the same topology ) on the entirety may be lost. However, applies to two equivalent norms on a normed space that exactly one is complete when the other it is. In the case of normalized areas completeness is therefore a property of the standard topology that does not depend on the particular standard.

Sets and properties

• A normed space is a Banach space if and only if it converges in every absolutely convergent series.

• Every normed space can be complete, thereby obtaining a Banach space that contains the original space as a dense subspace.

• Is a linear map between two normed spaces is an isomorphism, then it follows from the completeness of the completeness of.

• Every finite- normed space is a Banach space. Conversely, a Banach space which has a countable Hamel basis is finite. The latter is a Konzequenz from the Baire property of complete metric spaces.

• Set of the open mapping: A continuous linear map between two Banach spaces is surjective if and only if it is open. Is bijective and continuous, then the inverse mapping is also continuous. It follows that every bijective bounded linear operator between Banach spaces is an isomorphism.

• Set by the closed graph: The graph of a linear map between two Banach spaces is exactly then completed in the product, if the map is continuous.

• Is a closed subspace of a Banach space, then is also a Banach space with the norm.

• The first isomorphism theorem for Banach spaces: Is the image of a bounded linear mapping between two Banach spaces completed, then. This is around the concept of topological isomorphism, ie there exists a bijective linear mapping from to so that both are continuous.

• The direct sum of normed spaces is a Banach space if and only if each of the individual rooms is a Banach space.

• Every Banach space is a Fréchet space.

• For every separable Banach space there exists a closed subspace of such that is.

• Banach - Steinhaus: Is a family of continuous linear operators from a Banach space into a normed space, then it follows from the pointwise boundedness of the uniform boundedness.

• Banach - Alaoglu: The closed unit ball in the dual space of a Banach space is weak -* compact.

Linear operators

Are and be normed spaces over the same body, so the set of all continuous linear maps is denoted by.

In infinite-dimensional spaces linear mappings are not necessarily continuous.

Is a vector space and by

Is a norm on defined. If a Banach space, as well.

If a Banach space, so is a Banach algebra with the identity operator than one element; the multiplication operation is given by the composition of linear maps.

Dual space

If a normed space and the underlying body, then even a Banach space is also ( with the absolute value of the norm ), and you can the topological dual space (also continuous dual space ) defined by. He is usually a proper subspace of the algebraic dual space.

• If a normed space, so is a Banach space.

• Be a normed space. Is separable as well.

The topological dual space can be used to define a topology: the weak topology. The weak topology is not equivalent to the norm topology if the space is infinite dimensional. From the convergence of a sequence in the norm topology is always followed by the convergence in the weak topology, conversely, generally not. In this sense, the convergence condition, which results from the weak topology, " weaker ".

There is a natural map from to ( the Bidualraum ) defined by: for all and. It follows from the Hahn- Banach theorem that for each of the map is continuous and therefore an element of. The picture is always injective and continuous ( even isometric ).

Reflexivity

If the natural map also still surjective (and hence an isometric isomorphism ), so called the normed space is reflexive. There are the following relationships:

• Each reflexive normed space is a Banach space.

• A Banach space is reflexive if it is reflexive. Equivalent to this statement is that the unit ball of is compact in the weak topology.

• If a reflexive normed space is a Banach space and there exists a bounded linear operator from to, then it is reflexive.

• If a reflexive normed space. Then if and only separable if it is separable.

• Set of James For a Banach space are equivalent:   is reflexive.
• With so.

Tensor

Let and be two vector spaces. The tensor product of and is a vector space equipped with a bilinear map, which has the following universal property: If any bilinear map into a vector space, then there exists a unique linear map with.

There are several ways to define a norm on the tensor product of the underlying vector spaces, including the projective tensor product and the injective tensor product. The tensor product of complete spaces is not completely back in general. Therefore, it is understood in the theory of Banach spaces under a tensor product often its completion, this of course depends on the choice of the norm.

Examples

The following is the body or a compact Hausdorff space and a closed interval is. and are real numbers and. Next is a sigma- algebra, algebra, and a lot of a measure.

Position in the hierarchy of mathematical structures

Every Hilbert space is a Banach space, but not vice versa: However it can be on a Banach space if and only one compatible to standard scalar product defined, if in him the parallelogram law applies ( set of Jordan -von Neumann ).

Some important spaces in functional analysis, for example, the space of all infinitely differentiable functions or the space of all distributions on, are indeed complete, but no normed vector spaces and therefore no Banach spaces. In Fréchet spaces one has a complete metric, while LF- spaces are complete uniform vector spaces that appear as limiting cases of Fréchet spaces. These are special classes of locally convex spaces and topological vector spaces.

Every normed space can be uniquely complete up to isometric isomorphism, ie embed a dense subspace in a Banach space.

Fréchet derivative

It is possible to define the derivative of a function between two Banach spaces. Intuitively, one can see that, if an element of, the derivation of point in a continuous linear map, the approximated close in the order of distance.

It's called ( Fréchet ) differentiable at if a continuous linear mapping exists, so that

Applies. The limit here is taken over all sequences with non- zero element of which converge to 0. If the limit exists, it writes and calls it the ( Fréchet ) derivative of in. Further generalizations of the derivation follow by analogy with analysis on finite dimensional spaces. Common to all derivative terms, however, the question of the continuity of the linear mapping

This notion of derivation is a generalization of the usual derivative of functions since the linear maps are multiplications of real numbers on easy.

If differentiable at each point, then a further mapping between Banach (generally no linear diagram) and may again be differentiated, so that the higher derivatives are defined by. The -th derivative in point can thus be seen as a multi- linear map.

Differentiation is a linear operation in the following sense: Are and two figures which are differentiated, and are scalars, and, then is differentiable, and it is

The chain rule is also valid in this context. If one in and one is in differentiable function, then the composition is differentiable and the derivative is the composition of the derivatives

Also, directional derivatives can be extended to infinite-dimensional vector spaces, at this point, we refer to the Gâteaux differential.

Integration of Banach space - valued functions

Under certain conditions it is possible to integrate Banach space - valued functions. In the twentieth century, many different approaches have been presented to an integration theory of Banach space - valued functions. Examples are the Bochner integral, which Birkhoff integral and the Pettis integral. In finite-dimensional Banach spaces, these three different approaches lead to the integration ultimately to the same integral. For infinite-dimensional Banach spaces but this is generally not the case anymore.