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I am trying to undresteam Inverse Function Theorem so I googled and found a book

Calculus on Manifolds by Michael Spivak

I was happy because I have proof here but I stucked in first paragraph. Can anyone help me with this equation? I really dont get why it is equal. I tried to do this by definion of frechet's derivative but I got stuck in welter of symbols.

Here is the theorem : enter image description here And here is part of the proof wich i don't understeand (underlined) enter image description here Just why $D(\lambda^{-1})(f(a)) = \lambda^{-1}$ ?

For me it is just : $D(\lambda^{-1})(f(a)) = D((Df(a))^{-1})(f(a)) $

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Because in general, if $L: (E,\| \cdot \|) \to (F,\|\cdot \|)$ is a continuous linear map, then for any $x \in E$, $DL(x) = L$.

It is because:

$$\frac{1}{\|h\|}\| L(x+h) - L(x) - L(h)\| = 0 \to 0$$

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  • $\begingroup$ Thank you, thats 100% right and... easy! I will accept the answer as soon as I can (6 min more ). $\endgroup$ – MaLiN2223 Jan 25 '16 at 3:58

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