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?



constant rank theorem for banach spaces
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.


