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Probably doesn't deserve such specific (and dubious) attribution: the theorem is well-known in this form, and the proof is not difficult. →Banach spaces |
m Standard headings &/or gen fixes. using AWB |
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===Manifolds===
The inverse function theorem can be generalized to differentiable maps between [[differentiable manifold]]s. In this context the theorem states that for a differentiable map ''F'' : ''M''
:(d''F'')<sub>''p''</sub> : T<sub>''p''</sub>''M'' → T<sub>''F''(''p'')</sub>''N''
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===Banach spaces===
The inverse function theorem can also be generalized to differentiable maps between [[Banach space]]s. Let ''X'' and ''Y'' be Banach spaces and ''U'' an open neighbourhood of the origin in ''X''. Let ''F'' : ''U''
In the simple case where the function is a [[bijection]] between ''X'' and ''Y'', the function has a continuous inverse. This follows immediately from the [[open mapping theorem]].
==See
* [[Implicit function theorem]]
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