Oh...! Sometimes I can't see the obviousness. Thanks a lot for this, this was worthful. Thus now I must show whether or not $\displaystyle 3f(0)+f(1)>0$, knowing that f is a polynomial of degree $\displaystyle \leq 1$. If so, then it's false. As a counter example I can take $\displaystyle f(x)=-10x-10^9$. So the a) wouldn't be an inner product.

For the b), we have $\displaystyle f(0)^2+f(1)^2>0$ 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 $\displaystyle \langle f,g\rangle=\langle g,f\rangle$. It also works.

Another one would be $\displaystyle \langle af,g\rangle=a\langle f,g\rangle$ where a is a constant. It also works!!!

From wikipedia

Inner product space - Wikipedia, the free encyclopedia, the fourth condition would be $\displaystyle \langle f,f\rangle=0 \Rightarrow f=0$. Which works. So b) is an inner product.

Can you tell me if I used the good conditions?