- Banach manifold
In

mathematics , a**Banach manifold**is amanifold modeled onBanach spaces . Thus it is atopological space in which each point has a neighbourhoodhomeomorphic to anopen set in a Banach space (a more involved and formal definition is given below). Banach manifolds are one possibility of extending manifolds toinfinite dimension s.A further generalisation is to

Fréchet manifold s, replacing Banach spaces byFréchet space s. On the other hand, aHilbert manifold is a special case of a Banach manifold in which the manifold is locally modelled onHilbert space s.**Definition**Let "X" be a set. An

**atlas of class**"C"^{"r"}, "r" ≥ 0, on "X" is a collection of pairs (called**charts**) ("U"_{"i"}, "φ"_{"i"}), "i" ∈ "I", such that

# each "U"_{"i"}is asubset of "X" and the union of the "U"_{"i"}is the whole of "X";

# each "φ"_{"i"}is abijection from "U"_{"i"}onto anopen subset "φ"_{"i"}("U"_{"i"}) of some Banach space "E"_{"i"}, and for any "i" and "j", "φ"_{"i"}("U"_{"i"}∩ "U"_{"j"}) is open in "E"_{"i"};

# the crossover map::$varphi\_\{j\}\; circ\; varphi\_\{i\}^\{-1\}\; :\; varphi\_\{i\}\; (U\_\{i\}\; cap\; U\_\{j\})\; o\; varphi\_\{j\}\; (U\_\{i\}\; cap\; U\_\{j\})$

: is an "r"-times continuously differentiable function for every "i" and "j" in "I", i.e. the "r"

^{th}Fréchet derivative ::$mathrm\{d\}^\{r\}\; ig(\; varphi\_\{j\}\; circ\; varphi\_\{i\}^\{-1\}\; ig)\; :\; varphi\_\{i\}\; (U\_\{i\}\; cap\; U\_\{j\})\; o\; mathrm\{Lin\}\; ig(\; E\_\{i\}^\{r\};\; E\_\{j\}\; ig)$

: exists and is a continuous function with respect to the "E"

_{"i"}-normtopology on subsets of "E"_{"i"}and theoperator norm topology on Lin("E"_{"i"}^{"r"}; "E"_{"j"}.)One can then show that there is a unique

topology on "X" such that each "U"_{"i"}is open and each "φ"_{"i"}is a homeomorphism. Very often, this topological space is assumed to be aHausdorff space , but this is not necessary from the point of view of the formal definition.If all the Banach spaces "E"

_{"i"}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"_{"i"}be the same space, or evenisomorphic astopological vector space s. However, if two charts ("U"_{"i"}, "φ"_{"i"}) and ("U"_{"j"}, "φ"_{"j"}) are such that "U"_{"i"}and "U"_{"j"}have a non-empty intersection, a quick examination of the derivative of the crossover map:$varphi\_\{j\}\; circ\; varphi\_\{i\}^\{-1\}\; :\; varphi\_\{i\}\; (U\_\{i\}\; cap\; U\_\{j\})\; o\; varphi\_\{j\}\; (U\_\{i\}\; cap\; U\_\{j\})$

shows that "E"

_{"i"}and "E"_{"j"}must indeed be isomorphic as topological vector spaces. Furthermore, the set of points "x" ∈ "X" for which there is a chart ("U"_{"i"}, "φ"_{"i"}) with "x" in "U"_{"i"}and "E"_{"i"}isomorphic to a given Banach space "E" is both open and closed. Hence, one can without loss of generality assume that, on each 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"_{"i"}, "φ"_{"i"}) | "i" ∈ "I" } if the crossover map:$varphi\_\{i\}\; circ\; varphi^\{-1\}\; :\; varphi\; (U\; cap\; U\_\{i\})\; o\; varphi\_\{i\}\; (U\; cap\; U\_\{i\})$

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"

^{"r"}**-manifold**structure on "X" is then defined to be a choice of equivalence class of atlases on "X" of class "C"^{"r"}. If all the Banach spaces "E"_{"i"}are isomorphic as topological vector spaces (which is guaranteed to be the case if "X" is 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**^{"n"}, or even an open subset of**R**^{"n"}. 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, metric Banach manifold "X" can be 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-dimensionalFréchet 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

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