The problem of the minimum of a functional

In this article we study the problem of the minimum of a convex functional in a nonreflexive Banach space. We give a method for extending the original space and for extending the original functional onto the extended space in such a way that the problem of the minimum of the functional turns out to be solvable in the extended space and, moreover, that the lower bound of the functional on the original space and the lower bound of the extended functional coincide. Further, an abstract scheme is used to investigate the extension of a Sobolev space.