Extensions of Farkas-Minkowski's Lemma and Lemma and Stiemke's Lemma in the space l1

This paper extends Farkas-Mnkowski's Lemma and Stiemke's Lemma from the Euclidean space to (l
¹
, l
^∞
). The extensions of Farkas-Minkowski's Lemma and Stiemke's Lemma are the Basic Valuation Theorem in the case (l
¹
, l
^∞
). The security price is weakly arbitrage-free if and only if there exists a positive state vector; the security price is strictly arbitrage-free if and only if there exists a strictly positive state vector. The present value of the securities prices at date 0 is the value of their returns over all countably infinite possible states of nature at date 1.