I'm having some problem with applying the inverse function theorem to prove this. Let be a linear isomorphism, such that , and . I want to show is invertible in neighborhood of . Can someone help me with this?

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- May 26th 2011, 11:38 PMjackieInverse function theorem
I'm having some problem with applying the inverse function theorem to prove this. Let be a linear isomorphism, such that , and . I want to show is invertible in neighborhood of . Can someone help me with this?

- May 27th 2011, 02:19 AMgirdav
You have to show that the differential is invertible at . Since is linear, we have for all and and since , .

From the hypothesis, we have and using the definition of the differential we get that . We conclude that for all .