Essential Prerequisites for Competency in Calculus of Variations

It is not difficult to understand the need for and the objective of the calculus of variations. 

The classical brachistochrone problem and the isoperimetric problem associated with Dido of Carthage are cases in point.

However, the prerequisites for competency are not as apparent. 

A suggested list of pre-competencies are listed below.

An nonpareil familiarity with function spaces is essential to make the theoretical arguments stick. One can even say that function spaces are the biosphere in which the calculus of variations exists and flourishes.

If you don't live and breathe function spaces you may find the journey down the road of calculus of variations somewhat tough.

We can start with a simple example of C[0,1] as a warm-up example. This is the set of all functions defined and continuous on the closed interval bounded by 0 and 1.