1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook