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

1. Mar 15, 2013

ianchenmu

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

2. Relevant equations
n/a

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. Mar 15, 2013

Fredrik

Staff Emeritus
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