Prove Finite Dimensional Normed Vector Space is Differentiable

[cassiew]

Homework Statement

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).

Homework Equations

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 S_n=sum(from j=0 to n) of T^j converges. Moreover, I-T is invertible, (S_n) 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.

 

[ystael]

First try differentiating $$f$$ at $$I$$; your proposition 1 should help with this. Then see if you can find a way to transfer that computation to any $$T \in U$$.

 

[cassiew]

Nevermind, I think I figured it out.