Norm of a linear transformation

[CarmineCortez]

Homework Statement

||T|| = {max|T(x)| : |x|<=1} show this is equivalent to ||T|| = {max|T(x)| : |x| = 1}

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...

 

[radou]

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.

 

[ystael]

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

 

[CarmineCortez]

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.

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


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

 

[ystael]

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

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

 

[CarmineCortez]

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

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

 

[Landau]

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