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Parallelogram Equality+Inner Product Spaces

  1. Dec 21, 2004 #1
    It is true that if a norm satisfies the parallelogram equality then it must come from an inner product right (i.e. < , > is an inner product).? How in the world could you go about proving/showing this?
  2. jcsd
  3. Dec 21, 2004 #2
    ya that's true; here's how it's done:
    a) you need to know the parallelogram law (duh) but also the polarization identity: [tex] \| x+y\|^2 - \|x-y\|^2 = 4<x,y>[/tex]

    b) let V be a normed linear space in which the parallelgram law holds. Define <x,y> by the polarisation identity & prove that V with that inner product is an inner product space, and that [tex]\|x\| = \sqrt{<x,x>}[/tex]

    b*) see that spaces like [tex]l^\infty[/tex], [tex]l^1[/tex], C[a,b] with the uniform norm, [tex]c_0[/tex], don't satisfy the parallelogram law, and that there's no inner product (by a) ) that gives the norms for those spaces
  4. Dec 21, 2004 #3

    matt grime

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    You are of course not allowing fields of characteristic 2.
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