Introduction to Convex Analysis: The Hahn-Banach and Lax-Milgram Theorems

Before we start diving into integral functionals, it is important to devout some time to understand relevant facts for abstract variational problems. Because in such a situation we do not assume any explicit form of the underlying functional, these results cannot be as fine as those that can be shown when we materialize functionals and spaces. However, the general route to existence of minimizers is essentially the same for all kinds of functionals: it is called the direct methodDirect method of the Calculus of Variations. This chapter can be considered then as a brief introduction to the fundamental field of Convex Analysis.