Editing Inverse function theorem (section)

===Banach spaces===
The inverse function theorem can also be generalized to differentiable maps between [[Banach space]]s ''{{Mvar|X}}'' and ''{{Mvar|Y}}''.<ref>{{cite book |first=David G. |last=Luenberger |author-link=David Luenberger |title=Optimization by Vector Space Methods |location=New York |publisher=John Wiley & Sons |year=1969 |isbn=0-471-55359-X |pages=240–242 |url=https://books.google.com/books?id=lZU0CAH4RccC&pg=PA240 }}</ref> Let ''{{Mvar|U}}'' be an open neighbourhood of the origin in ''{{Mvar|X}}'' and <math>F: U \to Y\!</math> a continuously differentiable function, and assume that the Fréchet derivative <math>dF_0: X \to Y\!</math> of ''{{Mvar|F}}'' at 0 is a [[bounded linear map|bounded]] linear isomorphism of ''{{Mvar|X}}'' onto ''{{Mvar|Y}}''. Then there exists an open neighbourhood ''{{Mvar|V}}'' of <math>F(0)\!</math> in ''{{Mvar|Y}}'' and a continuously differentiable map <math>G: V \to X\!</math> such that <math>F(G(y)) = y</math> for all ''{{Mvar|y}}'' in ''{{Mvar|V}}''. Moreover, <math>G(y)\!</math> is the only sufficiently small solution ''{{Mvar|x}}'' of the equation <math>F(x) = y\!</math>.

There is also the inverse function theorem for [[Banach manifold]]s.<ref>{{cite book |first=Serge |last=Lang |author-link=Serge Lang |title=Differential Manifolds |location=New York |publisher=Springer |year=1985 |isbn=0-387-96113-5 |pages=13–19 }}</ref>