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Homework Help: Co-norm of an invertible linear transformation on R^n

  1. Mar 15, 2013 #1
    1. The problem statement, all variables and given/known data
    [itex]|\;|[/itex] is a norm on [itex]\mathbb{R}^n[/itex].
    Define the co-norm of the linear transformation [itex]T : \mathbb{R}^n\rightarrow\mathbb{R}^n[/itex] to be
    [itex]m(T)=inf\left \{ |T(x)| \;\;\;\; s.t.\;|x|=1 \right \}[/itex]
    Prove that if [itex]T[/itex] is invertible with inverse [itex]S[/itex] then [itex]m(T)=\frac{1}{||S||}[/itex].

    2. Relevant equations

    3. The attempt at a solution
    I think probably we need to do something with the norm, but I still can't get it... So thank you.
  2. jcsd
  3. Mar 15, 2013 #2


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    Equalities of the form inf X = A are often proved by showing that the two inequalities inf X ≤ A and inf X ≥ A both hold. Together they imply equality of course. One of these proofs will typically use that inf X a lower bound of X (consider an arbitrary member of X), and the other will typically use that inf is the greatest lower bound of the set.

    How is ##\|S\|## defined? Can you prove anything about the relationship between ##\|S\|## and ##\|T\|##?

    Edit: I have so far only proved the inequality ##m(T)\leq 1/\|S\|##. The idea that I think looks the most promising for a proof of the equality is to take a closer look at the set ##\{|Tx|:|x|=1\}##. What is its infimum? What is its supremum?
    Last edited: Mar 15, 2013
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