Let $\displaystyle G\subset L(\mathbb{R}^n;\mathbb{R}^n)$ be the subset of invertible linear transformations.

a) For $\displaystyle H\in L(\mathbb{R}^n;\mathbb{R}^n)$, prove that if $\displaystyle ||H||<1$, then the partial sum $\displaystyle L_n=\sum_{k=0}^{n}H^k$ converges to a limit $\displaystyle L$ and $\displaystyle ||L||\leq\frac{1}{1-||H||}$.

b) If $\displaystyle A\in L(\mathbb{R}^n;\mathbb{R}^n)$ satisfies $\displaystyle ||A-I||<1$, then $\displaystyle A$ is invertible and $\displaystyle A^{-1}=\sum_{k=0}^{\infty }H^k$ where $\displaystyle I-A=H$. (Hint: Show that $\displaystyle AL_n=H^{n+1}$)

c) Let $\displaystyle \varphi :G\rightarrow G$ be the inversion map $\displaystyle \varphi(A)=A^{-1}$. Prove that $\displaystyle \varphi$ is continuous at the identity $\displaystyle I$, using the previous two facts.

d) Let $\displaystyle A, C \in G$ and $\displaystyle B=A^{-1}$. We can write $\displaystyle C=A-K$ and $\displaystyle \varphi(A-K)=c^{-1}=A^{-1}(I-H)^{-1}$ where $\displaystyle H=BK$. Use this to prove that $\displaystyle \varphi$ is continuous at $\displaystyle A$.



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