Let $\mathbb R_d[t]$ be the set of univariate polynomials in the variable $t$ of degree $d$, and $S$ be the set of elements of $\mathbb R_d[t]$ that are nonnegative on $[0, 1]$. Does the following equivalent hold for a polynomial $p \in \mathbb R_d[t]$?
$$p \in S \iff \int_0^1 p(t) q(t) \ge 0 \; \forall q \in S.$$