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Prove Finite Dimensional Normed Vector Space is Differentiable

  1. Mar 2, 2010 #1
    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-1HT-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 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.

    3. 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.
  2. jcsd
  3. Mar 3, 2010 #2
    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].
  4. Mar 3, 2010 #3
    Nevermind, I think I figured it out.
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