SCHWEITZER 1984 Problem 7

Let $V$ be a finite-dimensional subspace of $C[0,1]$ such that every nonzero $f\in V$ attains positive value at some point. Prove that there exists a polynomial $P$ that is strictly positive on $[0,1]$ and orthogonal to $V$ , that is, for every $f \in V$ , $\int_{0}^{1} f(x)P(x)dx =0$ (F.39) [A. Pinkus, V. Totik]