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Math Help - norm for a linear transformation

  1. #1
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    norm for a linear transformation

    the norm for a linear transformation T on R^n is ||T|| = {max|T(x)| : |x|<= 1}

    i need to show this is equivalent to ||T|| = {max|T(x)| : |x| = 1}
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  2. #2
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    Quote Originally Posted by CarmineCortez View Post
    the norm for a linear transformation T on R^n is ||T|| = {max|T(x)| : |x|<= 1}

    i need to show this is equivalent to ||T|| = {max|T(x)| : |x| = 1}
    Let B= \{ x\in \mathbb{R} ^n : \| x \| \leq 1 \} and S=\{ x\in \mathbb{R} ^n : \| x \| =1 \} then \sup _{B} \{ \| T(x) \| \} = \sup _{B} \{ \| x \| \| T\left( \frac{x}{ \|x\| } \right) \| \} \leq \sup_{S} \{ \| T(y) \| \}

    where the last inequality follows since x\in B and \frac{x}{ \| x\| } \in S . The other inequality is immediate since S \subset B.

    PS. Notice that this proof only requires a normed vector space.
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