Let be the subset of invertible linear transformations.

a) For , prove that if , then the partial sum converges to a limit and .

b) If satisfies , then is invertible and where . (Hint: Show that )

c) Let be the inversion map . Prove that is continuous at the identity , using the previous two facts.

d) Let and . We can write and where . Use this to prove that is continuous at .

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