Let V be a finite dimensional normed vector space and let U= L(V)*, the set of invertible elements in L(V). Show, f:U-->U defined by f(T)= T-1 is differentiable at each T in U and moreover,
Df(T)H = -T-1HT-1
where Df(T)= f'(T).
Apparently these propositions are supposed to help (which I've already proved and can use):
1.) Let V be a normed vector space and suppose T is an element of L(V). If ||T||<1, then the sequence Sn=sum(from j=0 to n) of Tj converges. Moreover, I-T is invertible, (Sn) converges to (I-T)-1, and ||(I-T)-1||<= 1/(1-||T||).
2.) Suppose V is a normed vector space and T, an element of L(V), is invertible. If S is an element of L(V) and ||T-S||<||T-1||-1, then S is invertible.
The Attempt at a Solution
I've already proven that the function q:V-->V where q(T)=T2 is differentiable at each T and Dq(T)H= TH+HT, and I tried to prove this one using the same technique, but it's getting me nowhere.