MathDB
IMC 2008 Day 1 P2

Source:

July 26, 2014
vectoralgebrapolynomialIMCcollege contests

Problem Statement

Denote by V\mathbb{V} the real vector space of all real polynomials in one variable, and let γ:V→R\gamma :\mathbb{V}\to \mathbb{R} be a linear map. Suppose that for all f,g∈Vf,g\in \mathbb{V} with γ(fg)=0\gamma(fg)=0 we have γ(f)=0\gamma(f)=0 or γ(g)=0\gamma(g)=0. Prove that there exist c,x0∈Rc,x_0\in \mathbb{R} such that \gamma(f)=cf(x_0)  \forall f\in \mathbb{V}
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