Farkas’ Lemma in the bilinear setting and evaluation functionals

Abstract

AbstractWe prove the following Farkas’ Lemma for simultaneously diagonalizable bilinear forms: If $$A_1,\ldots ,A_k$$ A 1 , … , A k , and $$B:\mathbb {R}^n \times \mathbb {R}^n \rightarrow \mathbb {R}$$ B : R n × R n → R are bilinear forms, then one—and only one—of the following holds: $$B=a_1 A_1 + \cdots + a_k A_k,$$ B = a 1 A 1 + ⋯ + a k A k , with non-negative $$a_i\text {'s}$$ a i 's , there exists (x, y) for which $$A_1(x,y) \ge 0 , \ldots , A_k(x,y) \ge 0$$ A 1 ( x , y ) ≥ 0 , … , A k ( x , y ) ≥ 0 and $$B(x,y) < 0$$ 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.