A First Exposure to Functional Analysis

We focus on this chapter on the three main basic issues that are indispensable to tackle variational problems under a solid foundation, namely, From the very beginning, and as is usual in Mathematics, we will adopt an abstract viewpoint to gain as much generality as possible with the same effort. Fundamental spaces of functions and sequences will be explored as examples to illustrate concepts, fine points in results, counterexamples, etc. Spaces that are fundamental to Calculus of Variations and Partial Differential Equations (PDE, for short) will also be introduced as they will play a central role in our discussion. In particular, we will introduce the concept of weak derivative and weak solution of a PDE, and start studying the important Sobolev spaces.