Page 59 - Vector Analysis

§2.6 The Inverse Function Theorem 55

道首先應該要保留的條件是類似於微分不為零的這個條件。但是在多變數函數之下，微
分不為零的條件該怎麼呈現，這是第一個問題。而當我們觀察 (A.1)，應該可以猜出在多
變數版本裡面所該對應到的條件，即是 (Df )(x) 這個 bounded linear map 的可逆性。

另外，假設 f P C 1，那麼由 Theorem 1.87 我們知道在一個點 x0 如果 (Df )(x0) 可逆
的話，那麼在一個鄰域裡 (Df )(x) 都可逆。所以下面這個反函數定理的條件中只有 (Df )
在一個點可逆這個條件，因為我們暫時也只能討論在小區域的反函數存不存在。

Before proceeding, we first prove the following important proposition which is used
crucially in the proof of the inverse function theorem.

Proposition 2.60 (Contraction Mapping Principle). Let F Ď Rn be a closed subset (on
which every Cauchy sequence converges), and Φ : F Ñ F be a contraction mapping; that
is, there is a constant θ P [0, 1) such that

››Φ(x) ´ Φ(y)››Rn ď θ}x ´ y}Rn .
Then there exists a unique point x P F , called the fixed-point of Φ, such that Φ(x) = x.
Proof. Let x0 P F , and define xk+1 = Φ(xk) for all k P N Y t0u. Then

}xk+1 ´ xk}Rn = ››Φ(xk) ´ Φ(xk´1)››Rn ď θ}xk ´ xk´1}Rn ď ¨ ¨ ¨ ď θk}x1 ´ x0}Rn ;
thus if ℓ ą k,

}xℓ ´ xk}Rn ď }xk ´ xk+1}Rn + }xk+1 ´ xk+2}Rn + ¨ ¨ ¨ + }xℓ´1 ´ xℓ}Rn

ď (θk + θk+1 + ¨ ¨ ¨ + θℓ´1)}x1 ´ x0}Rn

ď θk(1 + θ + θ2 + ¨ ¨ ¨ )}x1 ´ x0}Rn = θk θ }x1 ´ x0}Rn . (2.8)
1´

Since θ P [0, 1), lim θk θ }x1 ´ x0}Rn = 0; thus
1´
kÑ8

@ ε ą 0, D N ą 0 Q }xk ´ xℓ}Rn ă ε @ k, ℓ ě N .

In other words, txku8k=1 is a Cauchy sequence in F . By assumption, xk Ñ x as k Ñ 8 for
some x P F . Finally, since Φ(xk) = xk+1 for all k P N, by the continuity of Φ we obtain that

Φ(x) = lim Φ(xk) = lim xk+1 = x
kÑ8 kÑ8

which guarantees the existence of a fixed-point.