Evaluation inner product question

Question

I know the evaluation inner product is a thing for $P_n$, but have a question about positive definiteness. Example...

$\langle f,g\rangle = f(x_1)\,g(x_1)+f(x_2)\,g(x_2)$ at $(3, 1)$

But if $f(x) = (x-3)(x-1)$, then $\langle f,f\rangle=0$. But $f$ is not the zero vector.

What am I missing?

Answer 1

Once $n$ is fixed, you need to evaluate at at least $n+1$ distinct points to get a positive definite scalar product on $P_n$, since a polynomial with degree $\leq n$ which is zero at $n+1$ points is necessarily $0$.