Farkas’ Lemma in the bilinear setting and evaluation functionals

Abstract

We prove the following Farkas’ Lemma for simultaneously diagonalizable bilinear forms: If A_1,ldots ,A_k, and B:mathbb {R}^n times mathbb {R}^n rightarrow mathbb {R} are bilinear forms, then one—and only one—of the following holds: B=a_1 A_1 + cdots + a_k A_k, with non-negative a_itext {'s},there exists (x, y) for which A_1(x,y) ge 0 , ldots , A_k(x,y) ge 0 and B(x,y) < 0. We study evaluation maps over the space of bilinear forms and consequently construct examples in which Farkas’ Lemma fails in the bilinear setting.