On inverse function theorem in Spivak's CoM

  • #1
psie
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TL;DR Summary
I have two questions on Spivak's proof of the inverse function theorem in Calculus on Manifolds.
I know of a thread on this site with a similar question, but no definite answer. I will not state the whole proof, as it is quite long.

2-11 Theorem (Inverse Function Theorem). Suppose that ##f: \mathbb{R}^n\to\mathbb{R}^n## 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\to W## has a continuous inverse ##f^{-1}:W\to 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^{-1}\circ f)(a) = D(\lambda^{-1})(f(a))\circ Df(a) = \lambda^{-1}\circ Df(a)## is the identity linear transformation. If the theorem is true for ##\lambda^{-1}\circ f##, it is clearly true for ##f##. Therefor we may assume at the outset that ##\lambda## is the identity.

... ... ...

Note that (3) and Lemma 2-10 applied to ##g(x)=f(x)-x## imply for ##x_1,x_2\in U## that $$|f(x_1)-x_1-(f(x_2)-x_2)|\leq\frac12|x_1-x_2|.$$ Since \begin{align*}
|x_1-x_2|-|f(x_1)-f(x_2)|&\leq |f(x_1)-x_1-(f(x_2)-x_2)| \\ &\leq\frac12 |x_1-x_2|,
\end{align*}
... ... ...

1. Why can we assume ##f## to have the identity map as its derivative? I understand how if the theorem is true for ##g = \lambda^{-1} \circ f##, then its true for ##f## and I also understand that ##Dg(a)=\mathrm{id}##, where ##\mathrm{id}## is the identity map. If ##\lambda=\mathrm{id}##, then simply ##g=f## and this is how the proof proceeds. But what when ##\lambda\neq \mathrm{id}##?

2. Why is $$|x_1-x_2|-|f(x_1)-f(x_2)|\leq |f(x_1)-x_1-(f(x_2)-x_2)|,$$ true? I'm tempted to say it is the reverse triangle inequality, but shouldn't there be an extra pair of absolute value signs around the LHS of this inequality?
 
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  • #2
Hi @psie,

1. Suppose the assertion holds true for ##g##. Then there are open neighborhoods ##V \ni a## and ##U\ni g(a)## such that ##g : V \to W## is a ##C^1##-diffeomorphism. Since ##\lambda : \mathbb{R}^n \to \mathbb{R}^n## is a ##C^1##-diffeomorphism, the image ##W := \lambda(U)## is an open neighborhood of ##f(a)## and the composition ##V \xrightarrow[\approx]{g} U \xrightarrow[\approx]{\lambda} W## is a ##C^1##-diffeomorphism. But this composition is ##f : V \to W##. The inverse ##f^{-1} : W \to V## is ##g^{-1} \circ \lambda^{-1}##, so the formula for ##(f^{-1})'(y)## is deduced from the formula for ##(g^{-1})'(z)## and the chain rule.

2. You are right that this is due to the reverse triangle inequality. Absolute value signs around the LHS are not needed since ##|x| \ge x## for all real numbers ##x##.
 
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  • #3
Thank you, @Euge. Regarding 1, I understand what you’re writing, but shouldn’t we then prove the theorem for ##g##? Spivak only seems to prove a very special case, namely when ##g=f##, i.e. when ##\lambda=\mathrm{id}##. How do we modify the proof when ##\lambda\neq\mathrm{id}##?
 
  • #4
If you multiply your function by a linear transformation the derivate is also multiplied by that linear transformation. So if you multiply it by the inverse of its derivative, you get a new function g whose derivative is the identity map.
 
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  • #5
psie said:
Thank you, @Euge. Regarding 1, I understand what you’re writing, but shouldn’t we then prove the theorem for ##g##?
That is what he did. Specifically, he proves the result when ##\lambda = \text{id}##. The general result is then obtained by considering ##\lambda^{-1} \circ f##. Since ##D(\lambda^{-1}\circ f)(a)## is the identity, by what he has shown, ##\lambda^{-1}\circ f## is ##C^1## locally invertible. Thus ##f = \lambda \circ (\lambda^{-1}\circ f)## is also ##C^1## locally invertible.
 
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