7. Let V be a finite-dimensional subspace of C[0,1] such that every nonzero f∈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∈V,∫01f(x)P(x)dx=0
(F.39)
[A. Pinkus, V. Totik] college contestsalgebrapolynomial