Obtaining variational formulations of elliptic boundary problems based on extensions of the classical theorems of Lax-Milgram (primal) or Babuška-Brezzi (mixed) to general contexts of locally convex spaces, as well as its discretisation, which allows the consideration of numerical schemes that make possible its resolution. In particular, it works with primal and mixed finite element methods and applications to various fields such as the elasticity are obtained