Is it an inner product ?

[Frandom]

Hello !

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 !

 

[Dr.Peterson]

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 ?

Is there a reason you are not certain? What doubts do you have?

Have you found a specific counterexample, or are you just supposing it must exist? To convince yourself, all you need to do is to state a proof, which in this case would be a counterexample.

It would be helpful if you showed the problem as given to you, and its context (e.g. what vector space are you working with, and what notation you are using), to help us interpret what you write more easily. I'd also like to see the definition you were given for an inner product, as it can be stated in slightly different ways. But these are all side issues.