Inverse Function Theorem in Spivak

[krcmd1]

In his proof of the IFT, on p. 36 of "Calculus on Manifolds," Spivak states: "If the theorem is true for \lambda^{-1} \circf, it is clearly true for f. Therefore we may assume at the outset that \lambda is the identity.

I don't understand why we may assume that.

thanks for your help!

Ken Cohen

 

[Office_Shredder]

For those of us without the textbook handy, can you post the context of what lambda is?

 

[krcmd1]

Is there a way to scan a page and post it?

 

[krcmd1]

"Suppose that f: R^{n} -> R^{m} is continuously differentiable in an open set containing a, and det f'(a) \neq 0. Then there is an open set V containing a and an open set W containing f(a) such that f: V -> W has a continuous inverse f^{-1}: W -> V which is differentiable and for all y \in W satisfies


(f^{-1})'(y) = [f'(f^{-1}(y))]^{-1}.

Proof. Let \lambda be the linear transformation Df(a). Then \lambda is non-singular, since det f'(a) \neq 0. Now D(\lambda\circf)(a) = D(\lambda^{-1})(f(a) = \lambda^{-1}\circDf(a) is the identity linear transformation."


This much I think I follow.

"If the theorem is true for \lambda^{-1}\circf, it is clearly true for f."

I think I understand this as well.

"Therefore we may assume at the ouset that \lambda is the identity"

That I don't understand. Since \lambda = Df(a), making it the identity seems a very severe condition on f(a).

It was easier that I thought to type this in with the Latex Reference. Thank you to whoever programmed that!

Ken Cohen