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| {{about|Banach bundles in differential geometry|Banach bundles in non-commutative geometry|Banach bundle (non-commutative geometry)}}
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| In [[mathematics]], a '''Banach bundle''' is a [[vector bundle]] each of whose fibres is a [[Banach space]], i.e. a [[complete metric space|complete]] [[normed vector space]], possibly of infinite dimension.
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| ==Definition of a Banach bundle==
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| Let ''M'' be a [[Banach manifold]] of class ''C''<sup>''p''</sup> with ''p'' ≥ 0, called the '''base space'''; let ''E'' be a [[topological space]], called the '''total space'''; let ''π'' : ''E'' → ''M'' be a [[surjective]] [[Continuous function (topology)|continuous map]]. Suppose that for each point ''x'' ∈ ''M'', the [[Fiber (mathematics)|fibre]] ''E''<sub>''x''</sub> = ''π''<sup>−1</sup>(''x'') has been given the structure of a Banach space. Let
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| :<math>\{ U_{i} | i \in I \}</math>
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| be an [[open cover]] of ''M''. Suppose also that for each ''i'' ∈ ''I'', there is a Banach space ''X''<sub>''i''</sub> and a map ''τ''<sub>''i''</sub>
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| :<math>\tau_{i} : \pi^{-1} (U_{i}) \to U_{i} \times X_{i}</math>
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| such that
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| * the map ''τ''<sub>''i''</sub> is a [[homeomorphism]] commuting with the projection onto ''U''<sub>''i''</sub>, i.e. the following [[commutative diagram|diagram commutes]]:
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| ::[[Image:CommDiag Local Triv Banach Bundle.png]]
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| : and for each ''x'' ∈ ''U''<sub>''i''</sub> the induced map ''τ''<sub>''ix''</sub> on the fibre ''E''<sub>''x''</sub>
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| ::<math>\tau_{ix} : \pi^{-1} (x) \to X_{i}</math>
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| : is an [[invertible function|invertible]] [[continuous linear map]], i.e. an [[isomorphism]] in the [[category (mathematics)|category]] of [[topological vector space]]s;
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| * if ''U''<sub>''i''</sub> and ''U''<sub>''j''</sub> are two members of the open cover, then the map
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| ::<math>U_{i} \cap U_{j} \to \mathrm{Lin}(X_{i}; X_{j})</math>
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| ::<math>x \mapsto (\tau_{j} \circ \tau_{i}^{-1})_{x}</math>
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| : is a [[morphism]] (a differentiable map of class ''C''<sup>''p''</sup>), where Lin(''X''; ''Y'') denotes the space of all continuous linear maps from a topological vector space ''X'' to another topological vector space ''Y''.
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| The collection {(''U''<sub>''i''</sub>, ''τ''<sub>''i''</sub>)|''i''∈''I''} is called a '''trivialising covering''' for ''π'' : ''E'' → ''M'', and the maps ''τ''<sub>''i''</sub> are called '''trivialising maps'''. Two trivialising coverings are said to be '''equivalent''' if their union again satisfies the two conditions above. An [[equivalence class]] of such trivialising coverings is said to determine the structure of a '''Banach bundle''' on ''π'' : ''E'' → ''M''.
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| If all the spaces ''X''<sub>''i''</sub> are isomorphic as topological vector spaces, then they can be assumed all to be equal to the same space ''X''. In this case, ''π'' : ''E'' → ''M'' is said to be a '''Banach bundle with fibre''' ''X''. If ''M'' is a [[connected space]] then this is necessarily the case, since the set of points ''x'' ∈ ''M'' for which there is a trivialising map
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| :<math>\tau_{ix} : \pi^{-1} (x) \to X</math>
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| for a given space ''X'' is both [[open set|open]] and [[closed set|closed]].
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| In the finite-dimensional case, the second condition above is implied by the first.
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| ==Examples of Banach bundles==
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| * If ''V'' is any Banach space, the [[tangent space]] T<sub>''x''</sub>''V'' to ''V'' at any point ''x'' ∈ ''V'' is isomorphic in an obvious way to ''V'' itself. The [[tangent bundle]] T''V'' of ''V'' is then a Banach bundle with the usual projection
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| ::<math>\pi : \mathrm{T} V \to V;</math>
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| ::<math>(x, v) \mapsto x.</math>
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| : This bundle is "trivial" in the sense that T''V'' admits a globally defined trivialising map: the [[identity function]]
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| ::<math>\tau = \mathrm{id} : \pi^{-1} (V) = \mathrm{T} V \to V \times V;</math>
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| ::<math>(x, v) \mapsto (x, v).</math>
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| * If ''M'' is any Banach manifold, the tangent bundle T''M'' of ''M'' forms a Banach bundle with respect to the usual projection, but it may not be trivial.
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| * Similarly, the [[cotangent bundle]] T*''M'', whose fibre over a point ''x'' ∈ ''M'' is the [[Dual space#Continuous dual space|topological dual space]] to the tangent space at ''x'':
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| ::<math>\pi^{-1} (x) = \mathrm{T}_{x}^{*} M = (\mathrm{T}_{x} M)^{*};</math>
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| : also forms a Banach bundle with respect to the usual projection onto ''M''.
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| * There is a connection between [[Bochner space]]s and Banach bundles. Consider, for example, the Bochner space ''X'' = ''L''²([0, ''T'']; ''H''<sup>1</sup>(Ω)), which might arise as a useful object when studying the [[heat equation]] on a domain Ω. One might seek solutions ''σ'' ∈ ''X'' to the heat equation; for each time ''t'', ''σ''(''t'') is a function in the [[Sobolev space]] ''H''<sup>1</sup>(Ω). One could also think of ''Y'' = [0, ''T''] × ''H''<sup>1</sup>(Ω), which as a [[Cartesian product]] also has the structure of a Banach bundle over the manifold [0, ''T''] with fibre ''H''<sup>1</sup>(Ω), in which case elements/solutions ''σ'' ∈ ''X'' are [[section (fiber bundle)|cross section]]s of the bundle ''Y'' of some specified regularity (''L''², in fact). If the differential geometry of the problem in question is particularly relevant, the Banach bundle point of view might be advantageous.
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| ==Morphisms of Banach bundles==
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| The collection of all Banach bundles can be made into a category by defining appropriate morphisms.
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| Let ''π'' : ''E'' → ''M'' and ''π''′ : ''E''′ → ''M''′ be two Banach bundles. A '''Banach bundle morphism''' from the first bundle to the second consists of a pair of morphisms
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| :<math>f_{0} : M \to M';</math>
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| :<math>f : E \to E'.</math>
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| For ''f'' to be a morphism means simply that ''f'' is a continuous map of topological spaces. If the manifolds ''M'' and ''M''′ are both of class ''C''<sup>''p''</sup>, then the requirement that ''f''<sub>0</sub> be a morphism is the requirement that it be a ''p''-times continuously [[differentiable function]]. These two morphisms are required to satisfy two conditions (again, the second one is redundant in the finite-dimensional case):
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| * the diagram
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| ::[[Image:CommDiag Banach Bundle Morphism.png]]
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| : commutes, and, for each ''x'' ∈ ''M'', the induced map
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| ::<math>f_{x} : E_{x} \to E'_{f_{0} (x)}</math>
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| : is a continuous linear map;
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| * for each ''x''<sub>0</sub> ∈ ''M'' there exist trivialising maps
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| ::<math>\tau : \pi^{-1} (U) \to U \times X</math>
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| ::<math>\tau' : \pi'^{-1} (U') \to U' \times X'</math>
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| : such that ''x''<sub>0</sub> ∈ ''U'', ''f''<sub>0</sub>(''x''<sub>0</sub>) ∈ ''U''′,
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| ::<math>f_{0} (U) \subseteq U'</math> | |
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| : and the map
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| ::<math>U \to \mathrm{Lin}(X; X')</math>
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| ::<math>x \mapsto \tau'_{f_{0} (x)} \circ f_{x} \circ \tau^{-1}</math>
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| : is a morphism (a differentiable map of class ''C''<sup>''p''</sup>).
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| ==Pull-back of a Banach bundle==
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| One can take a Banach bundle over one manifold and use the [[Pullback bundle|pull-back]] construction to define a new Banach bundle on a second manifold.
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| Specifically, let ''π'' : ''E'' → ''N'' be a Banach bundle and ''f'' : ''M'' → ''N'' a differentiable map (as usual, everything is ''C''<sup>''p''</sup>). Then the '''pull-back''' of ''π'' : ''E'' → ''N'' is the Banach bundle ''f''*''π'' : ''f''*''E'' → ''M'' satisfying the following properties:
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| * for each ''x'' ∈ ''M'', (''f''*''E'')<sub>''x''</sub> = ''E''<sub>''f''(''x'')</sub>;
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| * there is a commutative diagram
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| ::[[Image:CommDiag Pullback Banach Bundle 1.png]]
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| : with the top horizontal map being the identity on each fibre;
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| * if ''E'' is trivial, i.e. equal to ''N'' × ''X'' for some Banach space ''X'', then ''f''*''E'' is also trivial and equal to ''M'' × ''X'', and
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| ::<math>f^{*} \pi : f^{*} E = M \times X \to M</math>
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| : is the projection onto the first coordinate;
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| * if ''V'' is an open subset of ''N'' and ''U'' = ''f''<sup>−1</sup>(''V''), then
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| ::<math>f^{*} (E_{V}) = (f^{*} E)_{U}</math>
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| : and there is a commutative diagram
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| ::[[Image:CommDiag Pullback Banach Bundle 2.png]]
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| : where the maps at the "front" and "back" are the same as those in the previous diagram, and the maps from "back" to "front" are (induced by) the inclusions.
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| ==References== | |
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| * {{cite book | last=Lang | first=Serge | authorlink=Serge Lang | title=Differential manifolds | publisher=Addison-Wesley Publishing Co., Inc. | location=Reading, Mass.–London–Don Mills, Ont. | year=1972 }}
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| [[Category:Differential geometry]]
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| [[Category:Vector bundles]]
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