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

[ianchenmu]

Homework Statement

|\;| 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||}.

Homework Equations

n/a

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.

 

[Fredrik]

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?