ABSTRACT
This chapter presents some basic elements from various areas of linear algebra and functional analysis such as sets, function spaces, theory of functions, differential operators and calculus of variations. These elements provide the necessary concepts and basic principles for developing a mathematical framework and associated computational infrastructure for finite element processes. The objective here is to help readers refresh the material essential for the further development of the concepts and principles related to the mathematics of computations and finite element subject for BVPs. When the elements of a space are functions, we refer to such spaces as function spaces. In the theory of differential operators, function spaces contain desired functions. Functionals are variable quantities and play a very important role in mathematical physics, sciences, and engineering. Calculus of variations is a branch of applied mathematics that deals with extrema of functionals, that is, maximum, saddle points, and minimum.
