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#### mozganutyj

##### New member

- Jan 12, 2014

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I've encountered some problems while looking through the book called "Operator Algebras" by Bruce Blackadar.

At the very beginning there is a definition of pre-inner product on the complex vector space: briefly, it's the same as the inner product, but the necessity of x=0 when [x,x]=0 holds is omitted.

The point is that this stuff is followed by the script that "Cauchy-Schwarz inequality" holds for pre-inner product and here is the place I'm stuck in:

I can't prove the trivial case: [y,y]=0 implies [x,y]=0 for every x (otherwise the CBS inequality won't hold as the right-hand side equals to zero in this case - therefore, the only chance for the CBS inequality to be satisfied is [x,y]=0).

I suspect that CBS holds when we've got the inner product, not only the pre-inner.

I'll appreciate every effort of assistance from your side!