Special Properties of Hilbert Spaces?

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Hilbert spaces are distinguished from generic normed inner-product spaces primarily by their completeness; every Cauchy sequence in a Hilbert space converges within the space. The norm in a Hilbert space is generated by an inner product, which satisfies properties like the Polarization Identity and Parallelogram Law. This completeness allows for unique projections onto subspaces, which is crucial for the spectral theorem and the Riesz-Fischer representation theorem. While any metric space can be made complete, the specific properties of Hilbert spaces enable significant mathematical applications, particularly in functional analysis and probability theory. Understanding these distinctions is essential for leveraging the benefits of completeness in various mathematical contexts.
  • #31
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.
 
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  • #32
And the counter-example is a norm that actually does come from some inner product. Just rescale the inner product.
 
  • #33
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
O.K, good point.
 

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