A “from scratch” proof of a theorem of Rockafellar and Fulkerson

The theorem of this paper is of the same general class as Farkas' Lemma, Stiemke's Theorem, and the Kuhn—Fourier Theorem in the theory of linear inequalities. LetV be a vector subspace ofR ⁿ , and let intervalsI ¹,, I ⁿ of real numbers be prescribed. A necessary and sufficient condition is given for existence of a vector (x ¹ ,, x ⁿ ) inV such thatx ⁱ I ⁱ (i = 1, ,n); this condition involves the elementary vectors (nonzero vectors with minimal support) ofV . The proof of the theorem uses only elementary linear algebra.The author at present holds a Senior Scientist Award of the Alexander von Humboldt Stiftung.