Norm of a linear transformation

1. Jan 27, 2010

CarmineCortez

1. The problem statement, all variables and given/known data
||T|| = {max|T(x)| : |x|<=1} show this is equivalent to ||T|| = {max|T(x)| : |x| = 1}

3. The attempt at a solution

{max |T(x)| : x<=1} = {max ||x|| ||T(x/||x||)|| : |x|<=1} <= {max ||T(x)|| : |x| = 1}

does that look right? I need to show equality...

2. Jan 27, 2010

I assume you are speaking of a bounded linear transformation?

If T is bounded, then there exists some constant C so that ||Tx||<=C||x|| for all x from the domain of T, and it follows almost directly from this definition.

3. Jan 27, 2010

ystael

You're on the right track. What is the relation between the vector $$Tx$$ and the vector $$T\left(\frac{x}{\|x\|}\right)$$ ?

4. Jan 27, 2010

CarmineCortez

I don't know that T is bounded...T is on R^n

Tx >= T(x/||x||)

5. Jan 27, 2010

ystael

$$Tx$$ and $$T\left(\frac{x}{\|x\|}\right)$$ are vectors in the range space of $$T$$; they do not possess an order relation. The relationship between these two vectors is simpler, and follows directly from the definition of a linear transformation.

6. Jan 27, 2010

CarmineCortez

(1/||x|| ) Tx = T(x/||x||)

7. Jan 27, 2010

Landau

Exactly, by linearity! So, what can you conclude? Try to think a bit for yourself.