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Math Help - Different Forms of Operator Norm

  1. #1
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    Different Forms of Operator Norm



    I was given the very first inf definition, and have to prove that 3rd is the same as the given one. What I tried is taking sup on both sides of llAvll =< llAll llvll, but failed. Any help would be greatly appreciated.
    Thanks
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  2. #2
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    Let \|A\|_0=\sup\nolimits_{\|v\|=1}\|Av\| and let \|A\|_1=\inf\{c:\|Av\|\leq c\|v\|\textrm{ for all }v\in V\}.

    Consider any c such that \|Av\|\leq c\|v\| for all v\in V and choose any u\in V with \|u\|=1.

    Then \|Au\|\leq c\|u\|=c and so \sup\nolimits_{\|u\|=1}\|Au\|\leq c, i.e. \|A\|_0\leq c. Taking the inf over all such c, we deduce that \|A\|_0\leq \|A\|_1.


    If v\in V is non-zero, let u=v/\|v\|. Then \|u\|=1 and so \|Au\|\leq\|A\|_0.

    It follows by linearity that \|Av\|\leq\|A\|_0\|v\| for all non-zero v\in V and therefore for all v\in V, since it is clearly true when v=0.

    Hence \|Av\|\leq c\|v\| for all v\in V when c=\|A\|_0. By the definition of \|A\|_1, we see that \|A\|_1\leq\|A\|_0.


    Thus we have \|A\|_0=\|A\|_1.
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  3. #3
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    Thank you very much
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