# 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 infinite-dimensional 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

${\ displaystyle (X, \ | \ cdot \ |)}$,

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 . ${\ displaystyle X}$ ${\ displaystyle \ mathbb {K}}$ ${\ displaystyle \ | \ cdot \ |}$${\ displaystyle X}$ ${\ displaystyle d (x, y) = \ | xy \ |}$

## 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

• 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 .${\ displaystyle T \ colon X \ rightarrow Y}$${\ displaystyle X}$${\ displaystyle T (X)}$
• 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.${\ displaystyle M}$${\ displaystyle X}$${\ displaystyle M}$ ${\ displaystyle X / M}$${\ displaystyle \ | x + M \ | = \ inf \ limits _ {m \ in M} \ | x + m \ |}$
• The first isomorphism theorem for Banach spaces: If the image of a bounded linear mapping between two Banach spaces is closed , then . This is the concept of topological isomorphism, i. That is, there is a bijective linear mapping from to such that both and are continuous.${\ displaystyle T}$${\ displaystyle X / \ operatorname {ker} (T) \ cong T (X)}$${\ displaystyle L}$${\ displaystyle X / \ operatorname {ker} (T)}$${\ displaystyle T (X)}$${\ displaystyle L}$${\ displaystyle L ^ {- 1}}$
• The direct sum of normalized spaces is a Banach space if and only if each of the individual spaces is a Banach space.${\ displaystyle X_ {1} \ oplus \ cdots \ oplus X_ {n}}$${\ displaystyle X_ {j}}$
• 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 .${\ displaystyle T}$${\ displaystyle T}$ ${\ displaystyle T ^ {- 1}}$
• Theorem of the closed graph : The graph of a linear mapping between two Banach spaces is closed in the product if and only if the mapping is continuous.${\ displaystyle T \ colon X \ to Y}$${\ displaystyle X \ times Y}$
• For every separable Banach space there is a closed subspace of such that is.${\ displaystyle X}$${\ displaystyle M}$${\ displaystyle l ^ {1}}$${\ displaystyle X \ cong l ^ {1} / M}$

## Linear operators

Are and normed spaces over the same body , so the amount of all continuous - linear maps with designated. ${\ displaystyle X}$${\ displaystyle Y}$${\ displaystyle \ mathbb {K}}$${\ displaystyle \ mathbb {K}}$ ${\ displaystyle T \ colon X \ rightarrow Y}$${\ displaystyle B (X, Y)}$

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

${\ displaystyle B (X, Y)}$is a vector space and through ${\ displaystyle \ mathbb {K}}$

${\ displaystyle \ | T \ |: = \ mathrm {sup} \ {\ | Tx \ |: x \ in X {\ text {with}} \ | x \ | \ leq 1 \}}$

is a norm defined on. Is a Banach space, so too . ${\ displaystyle B (X, Y)}$${\ displaystyle Y}$${\ displaystyle B (X, Y)}$

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. ${\ displaystyle X}$${\ displaystyle B (X) = B (X, X)}$ ${\ displaystyle \ mathrm {id} _ {X}}$

## 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 . ${\ displaystyle X}$${\ displaystyle \ mathbb {K}}$${\ displaystyle \ mathbb {K}}$${\ displaystyle X '= B (X, \ mathbb {K})}$${\ displaystyle X ^ {*}}$

• If a normalized space is, then it is a Banach space.${\ displaystyle X}$${\ displaystyle X '}$
• Be a normalized space. Is separable so too .${\ displaystyle X}$${\ displaystyle X '}$ ${\ displaystyle X}$

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". ${\ displaystyle X}$${\ displaystyle X}$${\ displaystyle X}$

There is a natural mapping from to (the bidual space), defined by: for all and . From Hahn-Banach'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). ${\ displaystyle F}$${\ displaystyle X}$${\ displaystyle X '' = (X ')' = B (X ', \ mathbb {K})}$${\ displaystyle F \ colon X \ to X '', F (x) (f) = f (x)}$${\ displaystyle x \ in X}$${\ displaystyle f \ in X '}$${\ displaystyle x}$${\ displaystyle X}$${\ displaystyle F (x) \ colon X '\ to \ mathbb {K}}$${\ displaystyle X ''}$${\ displaystyle F}$

## Reflexivity

If the natural mapping is also surjective (and thus an isometric isomorphism), the normalized space is called reflexive . The following relationships apply: ${\ displaystyle F \ colon X \ to X ''}$${\ displaystyle X}$

• 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 .${\ displaystyle X}$${\ displaystyle X '}$${\ displaystyle X}$
• If a reflexive normalized space is a Banach space and if there is a bounded linear operator from to , then is reflexive.${\ displaystyle X}$${\ displaystyle Y}$${\ displaystyle X}$${\ displaystyle Y}$${\ displaystyle Y}$
• Is a reflexive standardized space. Then if and separable if is separable.${\ displaystyle X}$${\ displaystyle X}$${\ displaystyle X '}$
• James theorem For a Banach space are equivalent: ${\ displaystyle X}$
• ${\ displaystyle X}$ is reflexive.
• ${\ displaystyle \ forall f \ in X '\ \ exists x \ in X}$with so that .${\ displaystyle \ left \ | x \ right \ | \ leq 1}$${\ displaystyle f (x) = \ left \ | f \ right \ |}$

## Tensor product

Universal property of the 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 . ${\ displaystyle X}$${\ displaystyle Y}$${\ displaystyle \ mathbb {K}}$ ${\ displaystyle X \ otimes Y}$${\ displaystyle X}$${\ displaystyle Y}$${\ displaystyle \ mathbb {K}}$${\ displaystyle Z}$${\ displaystyle T \ colon X \ times Y \ rightarrow Z}$${\ displaystyle T '\ colon X \ times Y \ rightarrow Z'}$${\ displaystyle \ mathbb {K}}$${\ displaystyle Z '}$${\ displaystyle f \ colon Z \ rightarrow Z '}$${\ displaystyle T '= f \ circ T}$

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 σ-algebra , a set algebra and a measure . ${\ displaystyle \ mathbb {K}}$ ${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {C}}$${\ displaystyle X}$${\ displaystyle I = [a, b]}$${\ displaystyle p}$${\ displaystyle q}$${\ displaystyle 1 ${\ displaystyle {\ tfrac {1} {q}} + {\ tfrac {1} {p}} = 1}$${\ displaystyle \ Sigma}$${\ displaystyle \ Xi}$${\ displaystyle \ mu}$

designation Dual space reflexive weak
complete
standard Surname
${\ displaystyle \ mathbb {K} ^ {n}}$ ${\ displaystyle \ mathbb {K} ^ {n}}$ Yes Yes ${\ displaystyle \ | x \ | _ {2} = \ left (\ sum _ {i = 1} ^ {n} | x_ {i} | ^ {2} \ right) ^ {1/2}}$ Euclidean space
${\ displaystyle \ ell _ {n} ^ {p}}$ ${\ displaystyle \ ell _ {n} ^ {q}}$ Yes Yes ${\ displaystyle \ | x \ | _ {p} = \ left (\ sum _ {i = 1} ^ {n} | x_ {i} | ^ {p} \ right) ^ {1 / p}}$ Space of finite-dimensional vectors with the p -norm
${\ displaystyle \ ell _ {n} ^ {\ infty}}$ ${\ displaystyle \ ell _ {n} ^ {1}}$ Yes Yes ${\ displaystyle \ | x \ | _ {\ infty} = \ max _ {1 \ leq i \ leq n} | x_ {i} |}$ Space of finite-dimensional vectors with the maximum norm
${\ displaystyle \ ell ^ {p}}$ ${\ displaystyle \ ell ^ {q}}$ Yes Yes ${\ displaystyle \ | x \ | _ {p} = \ left (\ sum _ {i = 1} ^ {\ infty} | x_ {i} | ^ {p} \ right) ^ {1 / p}}$ Space of the sequences that can be summed up in the p th power
${\ displaystyle \ ell ^ {1}}$ ${\ displaystyle \ ell ^ {\ infty}}$ No Yes ${\ displaystyle \ | x \ | _ {1} = \ sum _ {i = 1} ^ {\ infty} | x_ {i} |}$ Space of consequences that can be summed up in terms of amount
${\ displaystyle \ ell ^ {\ infty}}$ ${\ displaystyle ba (2 ^ {\ mathbb {N}})}$ No No ${\ displaystyle \ | x \ | _ {\ infty} = \ sup _ {i} | x_ {i} |}$ Space of limited consequences
${\ displaystyle c}$ ${\ displaystyle \ ell ^ {1}}$ No No ${\ displaystyle \ | x \ | _ {\ infty} = \ sup _ {i} | x_ {i} |}$ Space of Convergent Consequences
${\ displaystyle c_ {0}}$ ${\ displaystyle \ ell ^ {1}}$ No No ${\ displaystyle \ | x \ | _ {\ infty} = \ sup _ {i} | x_ {i} |}$ Space of zero sequences ; isomorphic but not isometric too${\ displaystyle c}$
${\ displaystyle bv \,}$ ${\ displaystyle \ ell ^ {1} + \ mathbb {K}}$ No Yes ${\ displaystyle \ | x \ | _ {bv} = | x_ {1} | + \ sum _ {i = 1} ^ {\ infty} | x_ {i + 1} -x_ {i} |}$ Space of consequences of limited variation
${\ displaystyle bv_ {0}}$ ${\ displaystyle \ ell ^ {1}}$ No Yes ${\ displaystyle \ | x \ | _ {bv_ {0}} = \ sum _ {i = 1} ^ {\ infty} | x_ {i + 1} -x_ {i} |}$ Space of zero sequences of limited variation
${\ displaystyle bs}$ ${\ displaystyle ba (2 ^ {\ mathbb {N}})}$ No No ${\ displaystyle \ | x \ | _ {bs} = \ sup _ {n} \ left | \ sum _ {i = 1} ^ {n} x_ {i} \ right |}$ Limited sums space; isometric isomorphic to${\ displaystyle \ ell ^ {\ infty}}$
${\ displaystyle cs}$ ${\ displaystyle \ ell ^ {1}}$ No No ${\ displaystyle \ | x \ | _ {bs} = \ sup _ {n} \ left | \ sum _ {i = 1} ^ {n} x_ {i} \ right |}$ Space of convergent sums; closed subspace of ; isometric isomorphic to${\ displaystyle bs}$${\ displaystyle c}$
${\ displaystyle B (X, \ Xi)}$ ${\ displaystyle ba (\ Xi)}$ No No ${\ displaystyle \ | f \ | _ {\ infty} = \ sup _ {x \ in X} | f (x) |}$ Space of limited measurable functions${\ displaystyle \ Xi}$${\ displaystyle X}$
${\ displaystyle C (X)}$ ${\ displaystyle rca (\ Sigma)}$ No No ${\ displaystyle \ | f \ | _ {\ infty} = \ sup _ {x \ in X} \ left | f (x) \ right |}$ Space of continuous functions with Borel's σ-algebra${\ displaystyle X}$
${\ displaystyle ba (\ Xi)}$ ? No Yes ${\ displaystyle \ | \ mu \ | _ {ba} = \ sup _ {A \ in \ Sigma} | \ mu | (A)}$ Space of bounded finite-additive signed measure on${\ displaystyle \ Xi}$
${\ displaystyle ca (\ Sigma)}$ ? No Yes ${\ displaystyle \ | \ mu \ | _ {ba} = \ sup _ {A \ in \ Sigma} | \ mu | (A)}$ Space of -additive measures ; closed subspace of${\ displaystyle \ sigma}$ ${\ displaystyle ba (\ Sigma)}$
${\ displaystyle rca (\ Sigma)}$ ? No Yes ${\ displaystyle \ | \ mu \ | _ {ba} = \ sup _ {A \ in \ Sigma} | \ mu | (A)}$ Space of regular Borel measures ; closed subspace of${\ displaystyle ca (\ Sigma)}$
${\ displaystyle L ^ {p} (\ mu)}$ ${\ displaystyle L ^ {q} (\ mu)}$ Yes Yes ${\ displaystyle \ | f \ | _ {L ^ {p}} = \ left (\ int | f | ^ {p} \, d \ mu \ right) ^ {1 / p}}$ Space of the Lebesgue integrable functions in the p- th power
${\ displaystyle BV (I)}$ ? No Yes ${\ displaystyle \ | f \ | _ {BV} = \ lim _ {x \ to a ^ {+}} f (x) + V_ {f} (I)}$ Space of functions of limited total variation
${\ displaystyle NBV (I)}$ ? No Yes ${\ displaystyle \ | f \ | _ {BV} = V_ {f} (I)}$ Space of functions of limited total variation, the limit value of which vanishes at ${\ displaystyle a}$
${\ displaystyle AC (I)}$ ${\ displaystyle \ mathbb {K} + L ^ {\ infty} (I)}$ No Yes ${\ displaystyle \ | f \ | _ {BV} = \ lim _ {x \ to a ^ {+}} f (x) + V_ {f} (I)}$ Space of absolutely continuous functions ; isomorphic to Sobolev space ${\ displaystyle W ^ {1,1} (I)}$
${\ displaystyle C ^ {n} (I)}$ ${\ displaystyle rca (I)}$ No No ${\ displaystyle \ | f \ | _ {C ^ {n}} = \ sum _ {i = 0} ^ {n} \ sup _ {x \ in I} | f ^ {(i)} (x) | }$ Smooth Functions Room ; isomorphic to${\ displaystyle \ mathbb {R} ^ {n} \ oplus C (I)}$

## Classification in the hierarchy of mathematical structures

Overview of abstract spaces in mathematics. An arrow is to be understood as an implication, i.e. That is, the space at the beginning of the arrow is also a space at the end of the arrow.

Every Hilbert space is a Banach space, but not the other way around. According to Jordan-von 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 . ${\ displaystyle \ mathbb {R} \ rightarrow \ mathbb {R}}$${\ displaystyle \ mathbb {R}}$

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 . ${\ displaystyle f \ colon V \ to W}$${\ displaystyle x}$${\ displaystyle V}$${\ displaystyle f}$${\ displaystyle x}$${\ displaystyle f}$${\ displaystyle x}$${\ displaystyle \ vert h \ vert}$

One calls (Fréchet) -differentiable in if there is a continuous linear mapping such that ${\ displaystyle f}$ ${\ displaystyle x}$${\ displaystyle A \ colon V \ to W}$

${\ displaystyle \ lim _ {h \ to 0} {\ | f (x + h) -f (x) -A (h) \ | \ over \ | h \ |} = 0}$

applies. The limit value is formed here over all sequences with non-zero 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 finite-dimensional spaces. However, what is common to all derivation terms is the question of the continuity of the linear mapping${\ displaystyle V}$${\ displaystyle Df (x) = A}$${\ displaystyle f}$${\ displaystyle x}$${\ displaystyle Df (x)}$

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. ${\ displaystyle \ mathbb {R} \ to \ mathbb {R}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {R}}$

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 . ${\ displaystyle f}$${\ displaystyle x}$${\ displaystyle V}$${\ displaystyle Df \ colon V \ to L (V, W)}$${\ displaystyle f}$${\ displaystyle n}$${\ displaystyle x}$${\ displaystyle V_ {n} \ to W}$

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 ${\ displaystyle f}$${\ displaystyle g}$${\ displaystyle V \ to W}$${\ displaystyle x}$${\ displaystyle r}$${\ displaystyle s}$${\ displaystyle \ mathbb {K}}$${\ displaystyle rf + sg}$${\ displaystyle x}$

${\ displaystyle D (rf + sg) (x) = rD (f) (x) + sD (g) (x)}$.

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 ${\ displaystyle f \ colon V \ to W}$${\ displaystyle x \ in V}$${\ displaystyle g \ colon W \ to X}$${\ displaystyle f (x)}$${\ displaystyle g \ circ f}$${\ displaystyle x}$

${\ displaystyle D (g \ circ f) (x) = D (g) (f (x)) \ circ D (f) (x).}$

Directional derivations can also be extended to infinitely 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 to an integration theory of Banach space-valued functions were presented. Examples are the Bochner integral , the Birkhoff integral and the Pettis integral . In finite-dimensional Banach spaces, these three different approaches to integration ultimately lead to the same integral. For infinite-dimensional 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.uni-tuebingen.de/~deitmar/LEHRE/frueher/2011-12/FA/FA.pdf >
• Robert E. Megginson: An Introduction to Banach Space Theory . Springer-Verlag (1998), ISBN 0-387-98431-3
• 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 non-Archimedean Banach spaces . Rheinisch-Westfälische Technische Hochschule Aachen. 1976
• Joseph Diestel: Sequences and series in Banach spaces , Springer-Verlag (1984), ISBN 0-387-90859-5
• Nelson Dunford; Jacob T. Schwartz: Linear Operators, Part I, General Theory 1958