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 !

Solution 1:

This is only a partial answer to the question:

1. It is not in Lang's book.

2. 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).

3. 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.