Infinite dimensional constant rank theorem Suppose you have an analytic map $\phi : E \rightarrow \mathbb{C}^n$, where $E$ is a complex Banach space, and such that the rank of $D \phi$ is constant. Is it true then that the set $\phi^{-1}(\{0\})$ is a Banach submanifold of $E$ (with finite codimension, and tangent space equal to $\ker D \phi$ ?)
I guess the question amounts to using an infinite version of the constant rank theorem, but I couldn't find a reference for it.
Thank you very much in advance !
 A: This is only a partial answer to the question:


*

*It is not in Lang's book. 

*If you follow the (standard) proof of the constant rank theorem (CRT), then it goes through in the infinite-dimensional setting, provided that the following extra hypothesis is satisfied:
$$
K_x=Ker(D\phi(x)), x\in E, 
$$ 
varies smoothly with respect to $x$. This means that there exists a smooth map 
$$
h: E\to Gr^k(E), h(x)=K_x. 
$$ 
Here $Gr^k(E)$ is the Grassmannian of Banach subspaces of $E$ which have fixed codimension $k$ ($k=n-r$ is the corank of $D\phi(x)$, which is constant by assumption: since the rank $r$ of the derivative is supposed to be constant). 

*I suspect (but do not have an example of) that the CRT fails even for Hilbert spaces $E$ without the extra assumption explained in 2.    
A: As already pointed out in the comments, the fact that you are dealing with a finite dimensional target is enough so that a suitable version of the proof carries over. 
A citable reference for a suitable version of this theorem can be found for example as Theorem F in 
https://arxiv.org/pdf/1502.05795.pdf
