(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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^{-1}HT^{-1}

where Df(T)= f'(T).

2. Relevant 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.

3. The attempt at a solution

I've already proven that the function q:V-->V where q(T)=T^{2}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.

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# Homework Help: Prove Finite Dimensional Normed Vector Space is Differentiable

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