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Norm of a linear transformation

  1. Jan 27, 2010 #1
    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. jcsd
  3. Jan 27, 2010 #2


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    Homework Helper

    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.
  4. Jan 27, 2010 #3
    You're on the right track. What is the relation between the vector [tex]Tx[/tex] and the vector [tex]T\left(\frac{x}{\|x\|}\right)[/tex] ?
  5. Jan 27, 2010 #4
    I don't know that T is bounded...T is on R^n

    Tx >= T(x/||x||)
  6. Jan 27, 2010 #5
    [tex]Tx[/tex] and [tex]T\left(\frac{x}{\|x\|}\right)[/tex] are vectors in the range space of [tex]T[/tex]; 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.
  7. Jan 27, 2010 #6

    (1/||x|| ) Tx = T(x/||x||)
  8. Jan 27, 2010 #7


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    Science Advisor

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