This paper deals with the characterization, in terms of closedness of certain sets regarding other sets, of Farkas lemmas determining when the upperlevel set of a convex function $f$ contains a set of the form $C\cap \mathbb{A}^{-1}\left ( D\right ) $ , where $C$ and $D$ are convex sets (not necessarily cones) in locally convex spaces $X$ (with topological dual $X^{\prime }$ ) and $Y$ , respectively, while $\mathbb{A}$ is a continuous linear operator from $X$ to $Y$ . More in detail, each of the mentioned characterizations of Farkas type lemmas consists in the closedness of certain subset of either one of the “primal” spaces $X\times Y\times \mathbb{R}$ and $Y\times \mathbb{R}$ , or of the “dual” space $X^{\prime }\times \mathbb{R}$ , regarding some singleton set of the corresponding space. Moreover, the paper also provides an existence theorem for the feasible set $C\cap \mathbb{A}^{-1}\left ( D\right ) $ in terms of the closedness of certain subset of the dual space $X^{\prime }\times \mathbb{R}$ regarding the singleton set formed by the null element. These results are illustrated with significant applications to constrained convex minimization problems and to functional approximation by polynomials.
