Inverse function theorem: Difference between revisions

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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'' → ''Y'' be continuously differentiable and assume that the derivative (d''F'')₀ : ''X'' → ''Y'' of ''F'' at 0 is a [[bounded linear map|bounded]] linear isomorphism of ''X'' onto ''Y''. Then there exists an open neighbourhood ''V'' of ''F''(0) in ''Y'' and a continuously differentiable map ''G'' : ''V'' → ''X'' such that ''F''(''G''(''y'')) = ''y'' for all ''y'' in ''V''. Moreover, ''G''(''y'') is the only sufficiently small solution ''x'' of the equation ''F''(''x'') = ''y''. The first Banach space version of the inverse function theorem was proved by Lawrence Graves in 1950.

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]].

==References==