Generalized separation theorems and the Farkas' lemma

The main purpose of this paper is to extend the conventional separation theorems concerning the convex subset of Rm to generalized separation theorems concerning the convex subset of Rm×Sn. This is accomplished by the introduction of the generalized inner product in Rm×Sn. Then we derive the famous Farkas' lemma in nonlinear semidefinite programming, which may be very important for the analysis of the dual theory and optimality conditions of nonlinear semidefinite programming.