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

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

**Physics Forums | Science Articles, Homework Help, Discussion**