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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 !

Albert
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Mor
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    Presumably the answer is yes. I think Lang's Differentiable Manifolds book develops the entire theory from the point of view of Banach manifolds. – Jim Belk May 08 '13 at 22:08

2 Answers2

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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.

Moishe Kohan
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  • Looking at the proof I don't think the extra assumption is necessary at all. The proof goes through without any modifications as far as I can see. – scineram Oct 25 '14 at 21:40
  • the proof does not need this. what is important is that the image is complemented, which is true because it is finite dimensional – Albert Jan 26 '15 at 16:43
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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

Alexander Schmeding
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