= Solution
{c}
The prescribed form is a sum of products of real <polynomial evaluations>. It is a <symmetric bilinear form>, since each evaluation is a <linear map>. Also
$$
\langle f,f\rangle=\frac{f(-1)^2+f(0)^2+f(1)^2}{3}\geq0.
$$
Equality forces $f(-1)=f(0)=f(1)=0$. A nonzero <polynomial> of <degree of a polynomial> at most two cannot have three distinct <roots of a polynomial>, by the <root bound for a polynomial>. Thus equality holds only for the zero <polynomial>, establishing <positive definiteness>. Therefore \b[the form is a real <inner product>], and $V$ is an <inner product space>. This is a <discrete sampling inner product on polynomial spaces>.
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