Banach space
A Banach space (also Banach space , Banach space ) is a completely normalized vector space in mathematics . Banach spaces are among the central study objects of functional analysis . In particular, many infinitedimensional function spaces are Banach spaces. They are named after the mathematician Stefan Banach , who presented them together with Hans Hahn and Eduard Helly in 1920–1922 .
definition
A Banach space is a completely normalized space
 ,
that is, a vector space over the field of real or complex numbers with a norm , in which each Cauchy sequence of elements of converges in the metric induced by the norm .
Explanations
With metric spaces, completeness is a property of the metric, not of the topological space itself. Moving to an equivalent metric (that is, a metric that creates the same topology) can lose completeness. For two equivalent norms in a standardized space, on the other hand, one is complete if and only if the other is. In the case of standardized spaces, completeness is therefore a property of the standard topology that does not depend on the specific standard.
Sentences and properties
 A normalized space is a Banach space if and only if every absolutely convergent series converges in it.
 Every normalized space can be completed , whereby a Banach space is obtained that contains the original space as a dense subspace .
 If a linear mapping between two normalized spaces is an isomorphism , then the completeness of follows from the completeness of .
 Every finitely dimensional normed space is a Banach space. Conversely, a Banach space, a maximum countable Hamel base has, finite. The latter is a consequence of the Bairean property of complete metric spaces.
 If a closed subspace is a Banach space , then it is again a Banach space. The factor space with the norm is then also a Banach space.
 The first isomorphism theorem for Banach spaces: If the image of a bounded linear mapping between two Banach spaces is closed , then . This is the concept of topological isomorphism, i. That is, there is a bijective linear mapping from to such that both and are continuous.
 The direct sum of normalized spaces is a Banach space if and only if each of the individual spaces is a Banach space.
 BanachSteinhaus theorem : If a family of continuous, linear operators from a Banach space into a normalized space follows from the pointwise boundedness the uniform boundedness follows .
 Theorem of the open mapping : A continuous linear mapping between two Banach spaces is surjective if and only if it is open. If bijective and continuous, then the inverse mapping is also continuous. It follows that every bijective bounded linear operator between Banach spaces is an isomorphism .
 Theorem of the closed graph : The graph of a linear mapping between two Banach spaces is closed in the product if and only if the mapping is continuous.
 BanachAlaoglu theorem : The closed unit sphere in the dual space of a Banach space is weak  *  compact .
 For every separable Banach space there is a closed subspace of such that is.
 Every Banach room is a Fréchet room .
Linear operators
Are and normed spaces over the same body , so the amount of all continuous  linear maps with designated.
In infinitedimensional spaces, linear mappings are not necessarily continuous.
is a vector space and through
is a norm defined on. Is a Banach space, so too .
If a Banach space, then is a Banach algebra with the identical operator as a unit ; the multiplication operation is given by the composition of linear maps.
Dual space
If a normalized space and the underlying body is , then itself is also a Banach space (with the absolute value as the norm), and the topological dual space (also continuous dual space) can be defined by . It is usually a real subspace of the algebraic dual space .
 If a normalized space is, then it is a Banach space.
 Be a normalized space. Is separable so too .
The dual topological space can be used to define a topology on : the weak topology . The weak topology is not equivalent to the standard topology when the space is infinitely dimensional. The convergence of a sequence in the norm topology always results in the convergence in the weak topology, and vice versa in general not. In this sense, the convergence condition resulting from the weak topology is "weaker".
There is a natural mapping from to (the bidual space), defined by: for all and . From HahnBanach's theorem it follows that for each of them the mapping is continuous and therefore an element of . The mapping is always injective and continuous (even isometric).
Reflexivity
If the natural mapping is also surjective (and thus an isometric isomorphism), the normalized space is called reflexive . The following relationships apply:
 Every reflexive normed space is a Banach space.
 A Banach space is reflexive if and only if is reflexive. It is equivalent to this statement that the unit sphere of is compact in the weak topology .
 If a reflexive normalized space is a Banach space and if there is a bounded linear operator from to , then is reflexive.
 Is a reflexive standardized space. Then if and separable if is separable.

James theorem For a Banach space are equivalent:
 is reflexive.
 with so that .
Tensor product
Be and two vector spaces. The tensor product of and is a vector space provided with a bilinear mapping , which has the following universal property : If there is any bilinear mapping into a vector space , then there is exactly one linear mapping with .
There are various ways of defining 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 generally not complete again. Therefore, in the theory of Banach spaces, a tensor product is often understood to be its completion, which of course depends on the choice of the norm.
Examples
In the following, the body or , is a compact Hausdorff space and a closed interval. and are real numbers with and . Next is a σalgebra , a set algebra and a measure .
designation  Dual space  reflexive 
weak complete 
standard  Surname 

Yes  Yes  Euclidean space  
Yes  Yes  Space of finitedimensional vectors with the p norm  
Yes  Yes  Space of finitedimensional vectors with the maximum norm  
Yes  Yes  Space of the sequences that can be summed up in the p th power  
No  Yes  Space of consequences that can be summed up in terms of amount  
No  No  Space of limited consequences  
No  No  Space of Convergent Consequences  
No  No  Space of zero sequences ; isomorphic but not isometric too  
No  Yes  Space of consequences of limited variation  
No  Yes  Space of zero sequences of limited variation  
No  No  Limited sums space; isometric isomorphic to  
No  No  Space of convergent sums; closed subspace of ; isometric isomorphic to  
No  No  Space of limited measurable functions  
No  No  Space of continuous functions with Borel's σalgebra  
?  No  Yes  Space of bounded finiteadditive signed measure on  
?  No  Yes  Space of additive measures ; closed subspace of  
?  No  Yes  Space of regular Borel measures ; closed subspace of  
Yes  Yes  Space of the Lebesgue integrable functions in the p th power  
?  No  Yes  Space of functions of limited total variation  
?  No  Yes  Space of functions of limited total variation, the limit value of which vanishes at  
No  Yes  Space of absolutely continuous functions ; isomorphic to Sobolev space  
No  No  Smooth Functions Room ; isomorphic to 
Classification in the hierarchy of mathematical structures
Every Hilbert space is a Banach space, but not the other way around. According to Jordanvon Neumann's theorem, a scalar product compatible with the norm can be defined on a Banach space if and only if the parallelogram equation applies in it.
Some important spaces in functional analysis, for example the space of all infinitely often differentiable functions or the space of all distributions , are complete, but not standardized vector spaces and therefore not Banach spaces. In Fréchet spaces one still has a complete metric , while LF spaces are complete uniform vector spaces that appear as borderline cases of Fréchet spaces. These are special classes of locally convex spaces or topological vector spaces .
Every normalized space can be uniquely completed except for isometric isomorphism, that is, embedded as a dense subspace in a Banach space.
Fréchet derivation
It is possible to define the derivative of a function between two Banach spaces. You can intuitively see that if there is an element of , the derivative of at the point is a continuous linear mapping that approximates close to the order of the distance .
One calls (Fréchet) differentiable in if there is a continuous linear mapping such that
applies. The limit value is formed here over all sequences with nonzero elements that converge to 0. If the limit exists, it is written and called the ( Fréchet ) derivative of in . Further generalizations of the derivation result analogously to the analysis on finitedimensional spaces. However, what is common to all derivation terms is the question of the continuity of the linear mapping
This notion of derivative is a generalization of the ordinary derivative of functions , since the linear mappings from to are simply multiplications with real numbers.
If is differentiable at every point of , then another mapping between Banach spaces (generally no linear map!) And can possibly be differentiated again, thus the higher derivatives of be defined. The th derivative in the point can thus be seen as a multilinear mapping .
Differentiation is a linear operation in the following sense: If and are two mappings that are differentiable in, and are and scalars out , then is differentiable in and it holds
 .
The chain rule is also valid in this context. If there is one in and one in differentiable function, then the composition in is differentiable and the derivative is the composition of the derivatives
Directional derivations can also be extended to infinitely dimensional vector spaces, at this point we refer to the Gâteaux differential .
Integration of Banach spacevalued functions
Under certain conditions it is possible to integrate Banach spacevalued functions. In the twentieth century many different approaches to an integration theory of Banach spacevalued functions were presented. Examples are the Bochner integral , the Birkhoff integral and the Pettis integral . In finitedimensional Banach spaces, these three different approaches to integration ultimately lead to the same integral. For infinitedimensional Banach spaces, however, this is generally no longer the case. Furthermore, one can move from ordinary measures to vectorial measures , which take on their values in Banach spaces, and define an integral with respect to such measures.
literature
 Stefan Banach: Théorie des opérations linéaires . Warszawa 1932. (Monograph Matematyczne; 1) Zbl 0005.20901 (Kolmogoroff)
 Prof. Dr. A. Deitmar: Functional Analysis Script WS2011 / 12 < http://www.mathematik.unituebingen.de/~deitmar/LEHRE/frueher/201112/FA/FA.pdf >
 Robert E. Megginson: An Introduction to Banach Space Theory . SpringerVerlag (1998), ISBN 0387984313
 Bernard Beauzamy: Introduction to Banach Spaces and their Geometry . North Holland. 1986
 Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces . Springer publishing house. 2000
 Anton Willkomm: Dissertation: On the representation theory of topological groups in nonArchimedean Banach spaces . RheinischWestfälische Technische Hochschule Aachen. 1976
 Joseph Diestel: Sequences and series in Banach spaces , SpringerVerlag (1984), ISBN 0387908595
 Nelson Dunford; Jacob T. Schwartz: Linear Operators, Part I, General Theory 1958