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

1. each ${\displaystyle U_{i}}$  is a subset of ${\displaystyle X}$  and the union of the ${\displaystyle U_{i}}$  is the whole of ${\displaystyle X}$ ;
2. each ${\displaystyle \varphi _{i}}$  is a bijection from ${\displaystyle U_{i}}$  onto an open subset ${\displaystyle \varphi _{i}\left(U_{i}\right)}$  of some Banach space ${\displaystyle E_{i},}$  and for any indices ${\displaystyle i{\text{ and }}j,}$  ${\displaystyle \varphi _{i}\left(U_{i}\cap U_{j}\right)}$  is open in ${\displaystyle E_{i};}$ 
3. the crossover map
   $${\displaystyle \varphi _{j}\circ \varphi _{i}^{-1}:\varphi _{i}\left(U_{i}\cap U_{j}\right)\to \varphi _{j}\left(U_{i}\cap U_{j}\right)}$$

    
   is an ${\displaystyle r}$ -times continuously differentiable function for every ${\displaystyle i,j\in I;}$  that is, the ${\displaystyle r}$ th Fréchet derivative
   $${\displaystyle \mathrm {d} ^{r}\left(\varphi _{j}\circ \varphi _{i}^{-1}\right):\varphi _{i}\left(U_{i}\cap U_{j}\right)\to \mathrm {Lin} \left(E_{i}^{r};E_{j}\right)}$$

    
   exists and is a continuous function with respect to the ${\displaystyle E_{i}}$ -norm topology on subsets of ${\displaystyle E_{i}}$  and the operator norm topology on ${\displaystyle \operatorname {Lin} \left(E_{i}^{r};E_{j}\right).}$ 

One can then show that there is a unique topology on ${\displaystyle X}$  such that each ${\displaystyle U_{i}}$  is open and each ${\displaystyle \varphi _{i}}$  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 ${\displaystyle E_{i}}$  are equal to the same space ${\displaystyle E,}$  the atlas is called an ${\displaystyle E}$ -atlas. However, it is not a priori necessary that the Banach spaces ${\displaystyle E_{i}}$  be the same space, or even isomorphic as topological vector spaces. However, if two charts ${\displaystyle \left(U_{i},\varphi _{i}\right)}$  and ${\displaystyle \left(U_{j},\varphi _{j}\right)}$  are such that ${\displaystyle U_{i}}$  and ${\displaystyle U_{j}}$  have a non-empty intersection, a quick examination of the derivative of the crossover map

A new chart ${\displaystyle (U,\varphi )}$  is called compatible with a given atlas ${\displaystyle \left\{\left(U_{i},\varphi _{i}\right):i\in I\right\}}$  if the crossover map

A ${\displaystyle C^{r}}$ -manifold structure on ${\displaystyle X}$  is then defined to be a choice of equivalence class of atlases on ${\displaystyle X}$  of class ${\displaystyle C^{r}.}$  If all the Banach spaces ${\displaystyle E_{i}}$  are isomorphic as topological vector spaces (which is guaranteed to be the case if ${\displaystyle X}$  is connected), then an equivalent atlas can be found for which they are all equal to some Banach space ${\displaystyle E.}$  ${\displaystyle X}$  is then called an ${\displaystyle E}$ -manifold, or one says that ${\displaystyle X}$  is modeled on ${\displaystyle E.}$ 

Every Banach space can be canonically identified as a Banach manifold. If ${\displaystyle (X,\|\,\cdot \,\|)}$  is a Banach space, then ${\displaystyle X}$  is a Banach manifold with an atlas containing a single, globally-defined chart (the identity map).

Similarly, if ${\displaystyle U}$  is an open subset of some Banach space then ${\displaystyle U}$  is a Banach manifold. (See the classification theorem below.)

It is by no means true that a finite-dimensional manifold of dimension ${\displaystyle n}$  is globally homeomorphic to ${\displaystyle \mathbb {R} ^{n},}$  or even an open subset of ${\displaystyle \mathbb {R} ^{n}.}$  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^[1] states that every infinite-dimensional, separable, metric Banach manifold ${\displaystyle X}$  can be embedded as an open subset of the infinite-dimensional, separable Hilbert space, ${\displaystyle H}$  (up to linear isomorphism, there is only one such space, usually identified with ${\displaystyle \ell ^{2}}$ ). 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 ${\displaystyle X.}$  Thus, in the infinite-dimensional, separable, metric case, the "only" Banach manifolds are the open subsets of Hilbert space.

• Banach bundle – vector bundle whose fibres form Banach spacesPages displaying wikidata descriptions as a fallback
• Differentiation in Fréchet spaces
• Finsler manifold – smooth manifold equipped with a Minkowski functional at each tangent spacePages displaying wikidata descriptions as a fallback
• 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

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• 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.