Special Properties of Hilbert Spaces?

  • #26
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Ah, I see, some people define an inner-product space as one where the norm derives from the inner-product.
I think everyone just defines it as a space that has an inner product. If it has another norm, just replace it with the one from the inner product. So, I don't know that there's any difference in definitions here. Having an inner product gives you a norm. It might be a different norm than some other norm that you might put on the space. So, the space is going to be an inner product space with respect to one norm, and it might not be with respect to a different norm.

It's a strange statement to say that C[0,1] with the sup norm AND inner product given by the integral is not a Hilbert space. It IS a Hilbert space. It's just that it now has two different norms on it, instead of one. It's better to say that the sup norm doesn't come from an inner product. The space by itself without the inner product is something that gets turned into a Hilbert space or Banach space, maybe in more than one possible way. It's not Hilbert until AFTER you put the inner product on it. And it's not Banach until AFTER you put a norm on it. There could be more than one possible inner product or norm. Just as a set could become a group in many possible ways, unless it has a prime number of elements or something, so that we know that it's cyclic.
 
  • #27
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Well, you're right, I meant to say and should have said that there is no inner-product that gives rise to the sup norm on C[0,1]. Then (C[0,1], Sup Norm) is not a Hilbert space. I stand corrected.
I meant to say that in some spaces, like in the case of C[0,1], there is a "standard/canonical" inner-product and a "standard/canonical" norm, and the two do not satisfy the norm being generated by an inner-product. If there is neither a canonical norm nor inner-product, then every space with an inner-product defined on it could trivially be made into a Hilbert space. So we ultimately start with a norm and then decide if the space with that norm is a Hilbert space.
 
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  • #28
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One more thing; I think the Cauchy-Schwarz inequality applies only to norms that come from inner-products.
 
  • #29
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One more thing; I think the Cauchy-Schwarz inequality applies only to norms that come from inner-products.
Well, considering there is an inner product that explicitly appears in the inequality, I would guess you are probably right. :p
 
  • #30
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Of course, what I meant is that you do not use the norm associated with the inner-product, the inequality may not hold.
Note of course that I did not state that the result may not hold in non-Hilbert cases, not that the result holds for Hilbert spaces. If you have an argument to this effect, I would gladly see it, if it as simple as you say.
 
  • #31
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Oh, I see what you were getting at.

To get a counter-example, multiply the inner product norm by a small constant. It's still a norm. And note that if you inner product something with itself, you have equality in the Cauchy-Schwartz inequality. The shrunken square norm will be less than the regular square norm.
 
  • #32
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And the counter-example is a norm that actually does come from some inner product. Just rescale the inner product.
 
  • #33
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I should add that the same trick works for a norm that doesn't come from an inner product, if you want it to work for one of those. If it did obey Cauchy-Schwarz, you could re-scale it so that it didn't for any given inner product.
 
  • #34
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O.K, good point.
 

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