Where is Banach Lemma used?

Question

Banach's Lemma:

Let $C\in\mathbb R^{n\times n}$ with $|C|<1$, then $I+C$ is invertible and

$\dfrac{1}{1+|C|}<|(I+C)^{-1}|<\dfrac{1}{1-|C|}$

How can we prove it? And exactly i don't understand this lemma where can it be used?

Answer 1

Suppose $(C+I)x=0$ for some $x\in\mathbb R^n$. Then, $Cx = -x$. However, $\|C\|< 1$ implies $|Cx|<|x|$. Contradiction. Thus, $C+I$ is invertible.

Morevoer, $$\|(C+I)^{-1}(C+I)\| = 1\implies\|(C+I)^{-1}\|\ge \frac1{\|I+C\|}\ge \frac1{\|I\|+\|C\|}=\frac1{1+\|C\|}$$

Similarly, $(C+I)^{-1}C +(C+I)^{-1}= I$ implies $$\|(C+I)^{-1}\|=\|I-(C+I)^{-1}C\|\le \|I\|+\|(C+I)^{-1}\|\|C\|\implies \|(C+I)^{-1}\|\le \frac1{1-\|C\|}$$