I have the following theorem
Suppose $B$ a Banach space, $U\subset \mathbb{C}$ is open and $F:U \to B$ is holomorphic such that $\exists~M~\forall z\in U, \lVert F(z)^{-1}\rVert < M$. Then $z \to F(z)^{-1}$ is holomorphic in $U$.
I'm confused in two parts. First, since it says inverse, instead of Banach space, shouldn't it be a Banach algebra?
The second doubt is in the part, $\exists~M~\forall z\in U, \lVert F(z)^{-1}\rVert < M$. I have two interpretations,
- $\forall z \in U, F(z)$ is invertible and $\lVert F(z)^{-1} \rVert<M$.
This being the interpretation I was able to demonstrate the result. For it suffices to verify that the inversion in a Banach algebra is holomorphic when $\lVert F(z)^{-1} \rVert<M$.
And the second interpretation,
- $\forall z \in U,$ such that $F(z)$ is invertible $\exists~M$ so that $\lVert F(z)^{-1} \rVert<M$.
This interpretation is stronger than the other one, as here it is not asking for it to be invertible in all domain. I believe there is a similar result for complex variable functions, but I could neither prove nor think of an example. Could this result be true?