constant rank theorem for banach spaces Is there a similar statement to the constant rank theorem for finite dim real smooth manifolds which holds for a smooth map $F:B \rightarrow M$ where $B$ is an infinite (countable) dim Banach space and $M$ is a finite dim real smooth manifold?
 A: Yes, there is. The (Constant) Rank Theorem for Banach spaces is Theorem 2.5.15 of the book of R. Abraham, J.E. Marsden and T. Ratiu, Manifolds, Tensor Analysis and Applications (3rd. edition, Springer-Verlag, 2001). There is a demand that the image of $DF[u_0]$ and the kernel of $DF[u_0]$ are closed direct summands for the $u_0\in B$ around which the theorem holds. The first requirement is automatic if $M$ is finite-dimensional.
A: There is also a version of the constant rank theorem in Glöckner's paper "Fundamentals of submersions and immersions between infinite-dimensional manifolds" (Theorem F of 1) which works specifically if the target is a finite-dimensional manifold (and the source an arbitrary manifold modeled on a locally convex space). 
The advantage of having a finite-dimensional target is that one can circumvent most of the tedious assumptions one needs for the case of an infinite-dimensional target (i.e. the ones from the version of Abraham, Marsden and Ratiu).
In 1 you can also find some references to constant rank theorems between Banach spaces. They are given after Theorem F.
A: An improvement to the theorem from the book Manifolds, Tensor Analysis and Applications of R. Abraham, J.E. Marsden and T. Ratiu mentioned above was achieved by J. Blot The Rank Theorem in Infinite Dimension. The assumption from Abraham et al. is that for every $u$ in a neighborhood of $u_0$ the subspace $DF[u]$ is closed and $DF[u]|_{B_1}$ is an isomorphism thereon, where $B_1$ is a closed complement of the kernel $ B_2 = \ker DF[u_0]$.
It can actually be weekend to the assumption
$$
 \operatorname{Range}(DF[u]) \cap M_2 = \emptyset
$$
where $M_2$ is the closed complement to $M_1 = \operatorname{Range}(DF[u_0]) $. 
Note that there is no condition for $DF[u]|_{E_1}$ to be surjective required. Conversely, Abraham et al.'s requirement implies Blot's requirement: Assuming that the intersection is not empty for arbitrarily small neighborhoods, we find $u_n$'s, $y_n$'s and $x_n$'s with $u_n\to u_0$, $y_n \in \operatorname{Range}(DF[u_n]) \cap B_2\setminus\{0\} $, $x_n \in B_1$, and $DF[u_n](x_n) = y_n $. 
The set $\{x \in M_2 \mid |x| = 1 \}$ is closed. Hence it has positive Hausdorff distance from $M_1$. Hence it's impossible that the distance of $y_n $ to $M_1$ vanishes as $n\to \infty$. This gives a contradiction.
