Proving a function is not an inner product

Question

The function is like a dot product given is for polynomials $p(x)$ and $q(x)$: $\langle p,q\rangle =\sum^{n}_{i=0}p(x_i)q(x_i).$ First I had to prove this is an inner product for each $P_k$ where $k\leq n$ and it was pretty straightforward to prove community, linearity, and positive definiteness. I now need to prove it is not an inner product for $k>n$ and am pretty sure this has to do with showing one of those properties doesn't apply but I've tested it a bit and they all seem to work. Does anyone have a hint?

Answer 1

Consider the polynomial $p(x) = a_{n+1}x^{n+1} + \cdots + a_{k}x^{k}$ and its norm with this inner product.