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Question about Normed Linear Spaces

  1. Apr 19, 2012 #1
    Statement: V is a finite dimensional vector space with basis {ei} (i goes from 1 to n). V has a norm || || defined on it(not necessarily induced by an inner product). Let x=Ʃxiei belong to V. I want to show that ||x|| ≥ ||xiei|| for any fixed i.

    I'm not entirely sure this result is correct. But i remember seeing something similar in a text a while ago.
    I know all the properties of a norm but i'm not sure how to proceed. I don't know how the independence of the basis vectors will fit into the proof.
     
  2. jcsd
  3. Apr 19, 2012 #2

    LCKurtz

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    It isn't true. In ##R^2## consider the basis ##e_1=<1,0>,\ e_2=<-1,.1>##. Then let ##x=1e_1+1e_2=<0,.1>##. Then ##\|x\|=.1<1\|e_1\|##.
     
  4. Apr 19, 2012 #3
    Thank you!
     
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