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Math Help - product rule in norm space

  1. #1
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    product rule in norm space

    the product rule fn->f , gn->g implies fngn->fg true in the normed
    vector space (C[0,1],||.||) depends on the the norm||.||. Give a proof or a
    counterexample for (C[0,1],||.||infinite),(C[0,1].||.||1)

    HINTS:For
    counterexample , you may wish to examine the case f=g=0 and choose fn=gn for
    some piecewise linear functions.

    can someone help me to solve out this
    question????? many thanks

    what i did for ||.|| infinite is that ||fn||->||f||, ||gn||->||g|| ,then (||fn||-||f||)*(||gn||-||g||)->0 ,||g||(||fn||-||f||)->0,||f||(||gn||-||g||)->0
    then get (||fn||-||f||)*(||gn||-||g||)+||g||(||fn||-||f||)+||f||(||gn||-||g||)=||fn||*||gn||-||f||*||g||->0
    therefore ||fn||*||gn||->||f||*||g||
    for it is infinite so we get ||fn|||*||gn||=||fngn|| and ||f||*||g||=||fg|| ( by definition of norm) so ||fn*gn||->||fg||
    i dont know it is right or wrong
    and by the ||.||1 i have no idea to get the counterexample
    Last edited by cummings123321; November 17th 2012 at 06:34 AM.
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