An inequality for continuous linear functionals

Let n > 1 be an integer, f ∈ C n [ a , b ] , and A : C [ a , b ] → R a continuous linear functional which annihilates all polynomials of degree at most n − 1 . We give sharp inequalities of the form | A ( f ) | ≤ M k ‖ f ( k ) ‖ 2 , k = 2 , … , n .