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Prove Finite Dimensional Normed Vector Space is Differentiable

  • Thread starter cassiew
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  • #1
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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 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.
 

Answers and Replies

  • #2
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First try differentiating [tex]f[/tex] at [tex]I[/tex]; your proposition 1 should help with this. Then see if you can find a way to transfer that computation to any [tex]T \in U[/tex].
 
  • #3
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Nevermind, I think I figured it out.
 

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