Banach manifold

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 $X$;
2. 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 $i{\text{ and }}j,$ $\varphi _{i}\left(U_{i}\cap U_{j}\right)$ is open in $E_{i};$
3. the crossover map $$\varphi _{j}\circ \varphi _{i}^{-1}:\varphi _{i}\left(U_{i}\cap U_{j}\right)\to \varphi _{j}\left(U_{i}\cap U_{j}\right)$$ is an $r$-times continuously differentiable function for every $i,j\in I;$ that is, the $r$th Fréchet derivative $$\mathrm {d} ^{r}\left(\varphi _{j}\circ \varphi _{i}^{-1}\right):\varphi _{i}\left(U_{i}\cap U_{j}\right)\to \mathrm {Lin} \left(E_{i}^{r};E_{j}\right)$$ 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 $\operatorname {Lin} \left(E_{i}^{r};E_{j}\right).$

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 $$\varphi _{j}\circ \varphi _{i}^{-1}:\varphi _{i}\left(U_{i}\cap U_{j}\right)\to \varphi _{j}\left(U_{i}\cap U_{j}\right)$$ 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 $$\varphi _{i}\circ \varphi ^{-1}:\varphi \left(U\cap U_{i}\right)\to \varphi _{i}\left(U\cap U_{i}\right)$$ 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 $(X,\|\,\cdot \,\|)$ 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 $\mathbb {R} ^{n},$ or even an open subset of $\mathbb {R} ^{n}.$ 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, 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

• Banach bundle – vector bundle whose fibres form Banach spacesPages displaying wikidata descriptions as a fallback
• Differentiation in Fréchet spaces
• Finsler manifold – Generalization of Riemannian manifolds
• Fréchet manifold – topological space modeled on a Fréchet space in much the same way as a manifold is modeled on a Euclidean spacePages displaying wikidata descriptions as a fallback
• Global analysis – which uses Banach manifolds and other kinds of infinite-dimensional manifolds
• Hilbert manifold – Manifold modelled on Hilbert spaces

References

• Abraham, Ralph; Marsden, J. E.; Ratiu, Tudor (1988). Manifolds, Tensor Analysis, and Applications. New York: Springer. ISBN 0-387-96790-7.
• Anderson, R. D. (1969). "Strongly negligible sets in Fréchet manifolds" (PDF). Bulletin of the American Mathematical Society. 75 (1). American Mathematical Society (AMS): 64–67. doi:10.1090/s0002-9904-1969-12146-4. ISSN 0273-0979. S2CID 34049979.
• Anderson, R. D.; Schori, R. (1969). "Factors of infinite-dimensional manifolds" (PDF). Transactions of the American Mathematical Society. 142. American Mathematical Society (AMS): 315–330. doi:10.1090/s0002-9947-1969-0246327-5. ISSN 0002-9947.
• Henderson, David W. (1969). "Infinite-dimensional manifolds are open subsets of Hilbert space". Bull. Amer. Math. Soc. 75 (4): 759–762. doi:10.1090/S0002-9904-1969-12276-7. MR 0247634.
• Lang, Serge (1972). Differential manifolds. Reading, Mass.–London–Don Mills, Ont.: Addison-Wesley Publishing Co., Inc.
• Zeidler, Eberhard (1997). Nonlinear functional analysis and its Applications. Vol.4. Springer-Verlag New York Inc.