The book I am going through says this :(adsbygoogle = window.adsbygoogle || []).push({});

The below proposition is false for real inner product spaces. As an example, consider the operator T in R^2 that is a counter clockwise rotation of 90 degrees around the origin. Thus , T(x,y) = (-y,x). Obviously, Tv is orthogonal to v for every v in R^2, even though T is not 0.

Proposition : if V is a complex inner product space and T is an inner product space on V such that <Tv,v>=0 for all v in V, then T =0.

They have given a proof which describes <Tu,w> in the form <Tx,x> and hence subsequently which proves that <Tu,w>=0 for all u,w in V. This implies that T=0. ( taking w = Tu ).

My doubt is that why is the condition of orthogonality for <Tv,v> =0 not valid for complex inner product vector space. Been confusing me .

Thanks

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# If V is a complex inner product and T is an operator on V such that <Tv,v> = 0

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