Banach manifold explained

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 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 dimensions.

A further generalisation is to Fréchet manifolds, replacing Banach spaces by Fréchet spaces. On the other hand, a Hilbert manifold is a special case of a Banach manifold in which the manifold is locally modeled on Hilbert spaces.

Definition

Let

X be a set. An atlas of class C ^{r,} r \geq 0 , on X is a collection of pairs (called charts) \left(U _{i,} \varphi _{i\right),} i \in I, such that

1. each

U _{i} is a subset of X and the union of the U _{i} is the whole of X ;

1. each

\varphi _{i} is a bijection from U _{i} onto an open subset \varphi _{i\left(U} _{i\right)} of some Banach space E _{i,} and for any indices i and j, \varphi _{i\left(U} _{i} \cap U _{j\right)} is open in E _{i;}

1. the crossover map $$\backslash varphi\_j\; \backslash circ\; \backslash varphi\_i^\; :\; \backslash varphi\_i\backslash left(U\_i\; \backslash cap\; U\_j\backslash right)\; \backslash to\; \backslash varphi\_j\backslash left(U\_i\; \backslash cap\; U\_j\backslash right)$$ is an r -times continuously differentiable function for every

i, j \in I; that is, the r th Fréchet derivative $$\backslash mathrm^r\backslash left(\backslash varphi\_j\; \backslash circ\; \backslash varphi\_i^\backslash right)\; :\; \backslash varphi\_i\backslash left(U\_i\; \backslash cap\; U\_j\backslash right)\; \backslash to\; \backslash mathrm\backslash left(E\_i^r;\; E\_j\backslash right)$$ exists and is a continuous function with respect to the E _{i} -norm topology on subsets of E _{i} and the operator norm topology on \operatorname{Lin}\left(E ^{r;} _{i} E _{j\right).}

One can then show that there is a unique topology on

X such that each U _{i} is open and each \varphi _{i} 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 _{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 even isomorphic as topological vector spaces. However, if two charts \left(U _{i,} \varphi _{i\right)} and \left(U _{j,} \varphi _{j\right)} are such that U _{i} and U _{j} have a non-empty intersection, a quick examination of the derivative of the crossover map$$\backslash varphi\_j\; \backslash circ\; \backslash varphi\_i^\; :\; \backslash varphi\_i\backslash left(U\_i\; \backslash cap\; U\_j\backslash right)\; \backslash to\; \backslash varphi\_j\backslash left(U\_i\; \backslash cap\; U\_j\backslash right)$$shows that E _{i} and E _{j} must indeed be isomorphic as topological vector spaces. Furthermore, the set of points x \in X for which there is a chart \left(U _{i,} \varphi _{i\right)} 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, \varphi) is called compatible with a given atlas \left\{\left(U _{i,} \varphi _{i\right)} : i \in I\right\} if the crossover map$$\backslash varphi\_i\; \backslash circ\; \backslash varphi^\; :\; \backslash varphi\backslash left(U\; \backslash cap\; U\_i\backslash right)\; \backslash to\; \backslash varphi\_i\backslash left(U\; \backslash cap\; U\_i\backslash right)$$is an r -times continuously differentiable function for every i \in 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

Every Banach space can be canonically identified as a Banach manifold. 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 homeomorphic to \Reals ^{n,} or even an open subset of \Reals ^{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, usually identified with \ell ^{2} ). 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.

See also

• • • • • Global analysis – which uses Banach manifolds and other kinds of infinite-dimensional manifolds

References

• Book: Abraham. Ralph. Marsden. J. E.. Ratiu. Tudor. 1988. Manifolds, Tensor Analysis, and Applications. Springer. New York. 0-387-96790-7.
• Anderson. R. D.. Strongly negligible sets in Fréchet manifolds. Bulletin of the American Mathematical Society. American Mathematical Society (AMS). 75. 1. 1969. 0273-0979. 10.1090/s0002-9904-1969-12146-4. 64–67. 34049979 .
• Anderson. R. D.. Schori. R.. Factors of infinite-dimensional manifolds. Transactions of the American Mathematical Society. American Mathematical Society (AMS). 142. 1969. 0002-9947. 10.1090/s0002-9947-1969-0246327-5. 315–330.
• Henderson. David W.. 1969. Infinite-dimensional manifolds are open subsets of Hilbert space. Bull. Amer. Math. Soc.. 75. 759–762. 10.1090/S0002-9904-1969-12276-7. 0247634. 4. free.
• Book: Lang, Serge. Serge Lang

. Serge Lang. Differential manifolds. 1972. Addison-Wesley Publishing Co., Inc.. Reading, Mass.–London–Don Mills, Ont..

• Book: Zeidler, Eberhard. 1997. Nonlinear functional analysis and its Applications. Vol.4. Springer-Verlag New York Inc..