I consider the following function (u,v) I-->(1/2 . u1.v2 ; 1/4. u2.v2). We have u=(u1,u2) and v=(v1,v2) of course.

The function doesn't define an inner product on R2 because the property (v,v)=0 <=> v=(0,0) fails. Do you agree with that or am I wrong ?

Thank you !