Farkas’ Lemma in the bilinear setting and evaluation functionals

Abstract

We prove the following Farkas’ Lemma for simultaneously diagonalizable bilinear forms: If A1, . . . , Ak, and B : Rn × Rn → R are bilinear forms, then one—and only one—of the following holds: (i) B = a1A1 +· · ·+ak Ak , with non-negative ai ’s, (ii) there exists (x, y) for which A1(x, y) ≥ 0, . . . , Ak (x, y) ≥ 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.