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Homework Help: Equivalent Norms =(

  1. Jul 4, 2011 #1
    Prove that two norms ||.||1 and ||.||2 are equivalent if and only if there exist 2 constants c and k such that c*||x||1 <= ||x||2 <= k*||x||1 for all x in the concerned vector space V.

    Attempt-> Equivalence implies a ball in norm 1 admits a ball in norm 2 and vice versa. For normed linear spaces, I know that B(x,r) = x + r*B(0,1).

    So, a ball with respect to norm 1, B1(x,r), admits a ball in norm 2 with say radius 's'.

    Using the normed linear space property, I can conclude that for a vector 'y' in V, if ||y||2 < s
    then ||y||1 < r.

    I don't know where im going =(
     
  2. jcsd
  3. Jul 4, 2011 #2
    Hi Oster! :smile:

    What is your definition of equivalent norms?
     
  4. Jul 4, 2011 #3
    HI! I got it =D

    My definition was that for every open ball with respect to norm 1, there existed an open ball w.r.t norm 2 contained in it and vice versaaaaaaa!
     
  5. Jul 4, 2011 #4
    Converse was easy pffff.
     
  6. Jul 4, 2011 #5

    HallsofIvy

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    Science Advisor

    That is the same as saying that a sequence converges in one norm if and only if it converges in the other.
     
  7. Jul 4, 2011 #6

    Gib Z

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    Homework Helper

    In another thread last night you were studying the analogous result for equivalent metrics. This follows from that result, as the norm induces a metric.
     
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