Special Properties of Hilbert Spaces?

[homeomorphic]

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.

 

[homeomorphic]

And the counter-example is a norm that actually does come from some inner product. Just rescale the inner product.

 

[homeomorphic]

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.

 

[WWGD]

O.K, good point.