Convex sets and their separation— revision the Hahn-Banach theorem: geometrical and analytical forms; their first consequences in lcs and in normed spaces. Convex functions: effective domain; epigraph; lower semi-continuity; sufficient conditions for continuity. The Gamma(V) set. Conjugate (or polar) functions and convex duality. Subdifferentiability and relation with Gâteaux differentiability. Elements of sub differential calculus.