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 !
