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

a) For H\in 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\frac{1}{1-||H||}.

b) If A\in 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 \varphi :G\rightarrow G be the inversion map \varphi(A)=A^{-1}. Prove that \varphi 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 \varphi(A-K)=c^{-1}=A^{-1}(I-H)^{-1} where H=BK. Use this to prove that \varphi is continuous at A.



I have little ideas about these questions. What's your answers? Thank you!