Proofs about invertible linear functions

[ianchenmu]

Homework Statement

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 $A$.

Homework Equations

n/a

The Attempt at a Solution

a) is easy by using the geometric series. For c and d, what $\varphi$ is continuous at the identity $I$ and $\varphi$ is continuous at $A$ mean? What we need to prove?

 

[micromass]

Hint: Absolute convergence.