Normed linear space and inner product space

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  • #1
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Homework Statement


Not all normed linear spaces are inner product spaces. Give examples.


Homework Equations


all equations and conditions constructing inner and normed linear spaces.


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.
 

Answers and Replies

  • #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.
 
  • #3
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I know all the sufficient properties for an inner product space, but how can I find a suitable example for this particular problem?
 
  • #4
morphism
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Did you do what I suggested? What sufficient properties did you find?
 
  • #5
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ah, do you mean positive definiteness or parallelogram law?
 
  • #6
morphism
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The latter is very useful!
 
  • #7
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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...
 
  • #8
morphism
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Work with a specific example. Take R^2 for instance. What norms do you know on this space?
 
  • #9
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thanks for the post!
 

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