Let $G\\subset L(\\mathbb{R}^n;\\mathbb{R}^n)$ be the subset of invertible linea

Question

Let $Gsubset L(mathbb{R}^n;mathbb{R}^n)$ be the subset of invertible linear transformations. a) For $Hin L(mathbb{R}^n;mathbb{R}^n)$, prove that if $||H||<1$, then the partial sum $L_n=sum_{k=0}^{n}H^k$ converges to a limit $L$ and $||L||leqrac{1}{1-||H||}$. b) If $Ain L(mathbb{R}^n;mathbb{R}^n)$ satisfies $||A-I||<1$, then A is invertible and $A^{-1}=sum_{k=0}^{infty }H^k$ where $I-A=H$. (Hint: Show that $AL_n=H^{n+1}$) c) Let $arphi :G ightarrow G$ be the inversion map $arphi(A)=A^{-1}$. Prove that $arphi$ is continuous at the identity I, using the previous two facts. d) Let $A, C in G$ and $B=A^{-1}$. We can write $C=A-K$ and $arphi(A-K)=c^{-1}=A^{-1}(I-H)^{-1}$ where $H=BK$. Use this to prove that $arphi$ is continuous at $A$.

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