Is there a version of the implicit function theorem for holomorphic functions between complex Banach spaces? If yes, do you know any reference?
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1Possibly answered here: https://math.stackexchange.com/questions/4061834/regular-value-theorem-and-sards-theorem-for-banach-manifolds – Eric Towers Jun 14 '24 at 02:59
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The inverse and implicit function theorems generalize verbatim to complex Banach spaces; see my remarks here. Just look at the proof of the real version of the inverse/implicit function theorem via the fixed point approach, and note that it doesn’t matter whether the underlying field is $\Bbb{F}=\Bbb{R}$ or $\Bbb{F}=\Bbb{C}$. The strength of the theorem in the complex case comes from the fact that we have seemingly weaker hypothesis of only requiring once complex differentiability (not even continuous differentiability). – peek-a-boo Jun 16 '24 at 23:07