Essentials of Mathematical Methods in Science and Engineering, 2nd Edition

CHAPTER 1FUNCTIONAL ANALYSIS

Functional analysis is the branch of mathematics that deals with spaces of functions and the transformation properties of functions between function spaces in terms of operators. Since these operators could be differential or integral, it makes functional analysis extremely useful in the study of differential and integral equations. Since the function and space concepts could be used to represent many different things, functional analysis has found a wide range of applications in science and engineering. It is also at the very foundation of numerical simulation. The most rudimentary concept of functional analysis is the definition of a function, which is basically a rule or a mapping that relates the members of one set of objects to the members of another set. In this chapter, we discuss the basic properties of functions like continuity, limit, convergence, inverse, differentiation, integration, etc.

1.1 CONCEPT OF FUNCTION

We start with a quick review of the basic concepts of set theory. Let be a set of objects of any kind: points, numbers, functions, vectors, etc. When is an element of the set , we show it as . For finite sets, we may define by ...