Why is this definition of inner product violating the axioms?

Question

for all real polynomials , I defined inner product of two polys to be (f.g)=f(1)g(1).

It seems to me that it does not violate any of axioms.

If there is , can you tell me for what axioms are violated?

Answer 1

For example, the polynomial defined by $f(x)=x-1$ is not the zero polynomial, yet it has $f\cdot f=0$.