proof of inverse function theorem
Let and consider the mapping
If we have
Also notice that . In fact, given ,
So is a contraction mapping and hence by the contraction principle there exists one and only one solution to the equation
i.e. is the only point in such that .
Hence given any we can find which solves . Let us call the mapping which gives this solution, i.e.
We now want to study the differentiability of . Let be any point, take and so small that . Let and define .
First of all notice that being
On the other hand we know that is differentiable in that is we know that for all it holds
with . So we get
|Title||proof of inverse function theorem|
|Date of creation||2013-03-22 13:31:20|
|Last modified on||2013-03-22 13:31:20|
|Last modified by||paolini (1187)|