Oh...! Sometimes I can't see the obviousness. Thanks a lot for this, this was worthful. Thus now I must show whether or not

, knowing that f is a polynomial of degree

. If so, then it's false. As a counter example I can take

. So the a) wouldn't be an inner product.

For the b), we have

as you said, which is always true if f is different from 0.

Now I must check if it also agrees with the 3 other conditions.

There is the condition

. It also works.

Another one would be

where a is a constant. It also works!!!

From wikipedia

Inner product space - Wikipedia, the free encyclopedia, the fourth condition would be

. Which works. So b) is an inner product.

Can you tell me if I used the good conditions?