Banach manifold
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In [[mathematics]], a '''Banach manifold''' is a [[manifold]] modeled on [[Banach spaces]]. Thus it is a [[topological space]] in which each point has a [[Neighbourhood (mathematics)|neighbourhood]] [[homeomorphic]] to an [[open set]] in a Banach space (a more involved and formal definition is given below). Banach manifolds are one possibility of extending manifolds to [[infinite]] [[dimension]]s.
A further generalisation is to [[Fréchet manifold]]s, replacing Banach spaces by [[Fréchet space]]s. On the other hand, a [[Hilbert manifold]] is a special case of a Banach manifold in which the manifold is locally modelled on [[Hilbert space]]s.
==Definition==
Let ''X'' be a [[set]]. An '''[[Atlas (topology)|atlas]] of class''' ''C''<sup>''r''</sup>, ''r'' ≥ 0, on ''X'' is a collection of pairs (called '''charts''') (''U''<sub>''i''</sub>, ''φ''<sub>''i''</sub>), ''i'' ∈ ''I'', such that
# each ''U''<sub>''i''</sub> is a [[subset]] of ''X'' and the [[union (set theory)|union]] of the ''U''<sub>''i''</sub> is the whole of ''X'';
# each ''φ''<sub>''i''</sub> is a [[bijection]] from ''U''<sub>''i''</sub> onto an [[open subset]] ''φ''<sub>''i''</sub>(''U''<sub>''i''</sub>) of some Banach space ''E''<sub>''i''</sub>, and for any ''i'' and ''j'', ''φ''<sub>''i''</sub>(''U''<sub>''i''</sub> ∩ ''U''<sub>''j''</sub>) is open in ''E''<sub>''i''</sub>;
# the crossover map
::<math>\varphi_{j} \circ \varphi_{i}^{-1} : \varphi_{i} (U_{i} \cap U_{j}) \to \varphi_{j} (U_{i} \cap U_{j})</math>
: is an [[Smooth function|''r''-times continuously differentiable]] function for every ''i'' and ''j'' in ''I'', i.e. the ''r''<sup>th</sup> [[Fréchet derivative]]
::<math>\mathrm{d}^{r} \big( \varphi_{j} \circ \varphi_{i}^{-1} \big) : \varphi_{i} (U_{i} \cap U_{j}) \to \mathrm{Lin} \big( E_{i}^{r}; E_{j} \big) </math>
: exists and is a continuous function with respect to the ''E''<sub>''i''</sub>-[[norm (mathematics)|norm]] [[topology]] on subsets of ''E''<sub>''i''</sub> and the [[operator norm]] topology on Lin(''E''<sub>''i''</sub><sup>''r''</sup>; ''E''<sub>''j''</sub>.)
One can then show that there is a unique [[topology]] on ''X'' such that each ''U''<sub>''i''</sub> is open and each ''φ''<sub>''i''</sub> is a homeomorphism. Very often, this topological space is assumed to be a [[Hausdorff space]], but this is not necessary from the point of view of the formal definition.
If all the Banach spaces ''E''<sub>''i''</sub> are equal to the same space ''E'', the atlas is called an ''E'''''-atlas'''. However, it is not ''[[a priori]]'' necessary that the Banach spaces ''E''<sub>''i''</sub> be the same space, or even [[isomorphic]] as [[topological vector space]]s. However, if two charts (''U''<sub>''i''</sub>, ''φ''<sub>''i''</sub>) and (''U''<sub>''j''</sub>, ''φ''<sub>''j''</sub>) are such that ''U''<sub>''i''</sub> and ''U''<sub>''j''</sub> have a non-empty [[intersection (set theory)|intersection]], a quick examination of the [[derivative (generalizations)|derivative]] of the crossover map
:<math>\varphi_{j} \circ \varphi_{i}^{-1} : \varphi_{i} (U_{i} \cap U_{j}) \to \varphi_{j} (U_{i} \cap U_{j})</math>
shows that ''E''<sub>''i''</sub> and ''E''<sub>''j''</sub> must indeed be isomorphic as topological vector spaces. Furthermore, the set of points ''x'' ∈ ''X'' for which there is a chart (''U''<sub>''i''</sub>, ''φ''<sub>''i''</sub>) with ''x'' in ''U''<sub>''i''</sub> and ''E''<sub>''i''</sub> isomorphic to a given Banach space ''E'' is both open and [[closed subset|closed]]. Hence, one can without loss of generality assume that, on each [[connected space|connected component]] of ''X'', the atlas is an ''E''-atlas for some fixed ''E''.
A new chart (''U'', ''φ'') is called '''compatible''' with a given atlas { (''U''<sub>''i''</sub>, ''φ''<sub>''i''</sub>) | ''i'' ∈ ''I'' } if the crossover map
:<math>\varphi_{i} \circ \varphi^{-1} : \varphi (U \cap U_{i}) \to \varphi_{i} (U \cap U_{i})</math>
is an ''r''-times continuously differentiable function for every ''i'' ∈ ''I''. Two atlases are called compatible if every chart in one is compatible with the other atlas. Compatibility defines an [[equivalence relation]] on the class of all possible atlases on ''X''.
A ''C''<sup>''r''</sup>'''-manifold''' structure on ''X'' is then defined to be a choice of equivalence class of atlases on ''X'' of class ''C''<sup>''r''</sup>. If all the Banach spaces ''E''<sub>''i''</sub> are isomorphic as topological vector spaces (which is guaranteed to be the case if ''X'' is [[connected space|connected]]), then an equivalent atlas can be found for which they are all equal to some Banach space ''E''. ''X'' is then called an ''E'''''-manifold''', or one says that ''X'' is '''modeled''' on ''E''.
==Examples==
* If (''X'', || ||) is a Banach space, then ''X'' is a Banach manifold with an atlas containing a single, globally-defined chart (the [[identity map]]).
* Similarly, if ''U'' is an open subset of some Banach space, then ''U'' is a Banach manifold. (See the classification theorem below.)
==Classification up to homeomorphism==
It is by no means true that a finite-dimensional manifold of dimension ''n'' is ''globally'' homeomorphic to '''R'''<sup>''n''</sup>, or even an open subset of '''R'''<sup>''n''</sup>. However, in an infinite-dimensional setting, it is possible to classify “[[well-behaved]]” Banach manifolds up to homeomorphism quite nicely. A 1969 theorem of David Henderson states that every infinite-dimensional, [[separable space|separable]], [[metric space|metric]] Banach manifold ''X'' can be [[embedding|embedded]] as an open subset of the infinite-dimensional, separable Hilbert space, ''H'' (up to linear isomorphism, there is only one such space). In fact, Henderson's result is stronger: the same conclusion holds for any metric manifold modeled on a separable infinite-dimensional [[Fréchet space]].
The embedding homeomorphism can be used as a global chart for ''X''. Thus, in the infinite-dimensional, separable, metric case, the “only” Banach manifolds are the open subsets of Hilbert space.
==References==
* {{cite journal
| last = Henderson
| first = David W.
| title = Infinite-dimensional manifolds are open subsets of Hilbert space
| journal = Bull. Amer. Math. Soc.
| volume = 75
| year = 1969
| pages = 759–762
| doi = 10.1090/S0002-9904-1969-12276-7
}} {{MathSciNet|id=0247634}}
* {{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
}}
[[es:variedad de Banach]]
[[Category:Differential geometry]]
[[Category:Structures on manifolds]]
[[Category:Nonlinear functional analysis]]