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Normed linear space and inner product space

  1. Oct 3, 2008 #1
    1. The problem statement, all variables and given/known data
    Not all normed linear spaces are inner product spaces. Give examples.


    2. Relevant equations
    all equations and conditions constructing inner and normed linear spaces.


    3. The attempt at a solution
    Well, I tried some of spaces like L space, but I didn't find any logical solutions. Frankly, I don't know from where to begin to prove this.
     
  2. jcsd
  3. Oct 3, 2008 #2

    morphism

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    Try to flip through your book to find a property inner product spaces satisfy that is stated in terms of the norm generated by the inner product.

    If you want another hint, post back.
     
  4. Oct 3, 2008 #3
    I know all the sufficient properties for an inner product space, but how can I find a suitable example for this particular problem?
     
  5. Oct 3, 2008 #4

    morphism

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    Did you do what I suggested? What sufficient properties did you find?
     
  6. Oct 3, 2008 #5
    ah, do you mean positive definiteness or parallelogram law?
     
  7. Oct 3, 2008 #6

    morphism

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    The latter is very useful!
     
  8. Oct 3, 2008 #7
    yes,ok, but how should I write down the associated norm of a space which I don't know.

    the only thing that I can write down is the parallelogram law itself which automatically yields the same thing in terms of inner product of its associated norm.

    I am a bit confused...
     
  9. Oct 3, 2008 #8

    morphism

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    Work with a specific example. Take R^2 for instance. What norms do you know on this space?
     
  10. Oct 3, 2008 #9
    thanks for the post!
     
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