In this chapter we recapitulate the mathematical preliminaries that will be relevant to the development of functional analysis in later chapters. This chapter comprises six sections. We presume that the reader has been exposed to an elementary course in real analysis and linear algebra.

Set

The theory of sets is one of the principal tools of mathematics. One type of study of set theory addresses the realm of logic, philosophy and foundations of mathematics. The other study goes into the highlands of mathematics, where set theory is used as a medium of expression for various concepts in mathematics. We assume that the sets are ‘not too big’ to avoid any unnecessary contradiction. In this connection one can recall the famous ‘Russell's Paradox’ (Russell, 1959). A set is a collection of distinct and distinguishable objects. The objects that belong to a set are called elements, members or points of the set. If an object a belongs to a set A, then we write a ∈ A. On the other hand, if a does not belong to A, we write a ∉ A. A set may be described by listing the elements and enclosing them in braces. For example, the set A formed out of the letters a, a, a, b, b, c can be expressed as A = {a, b, c}. A set can also be described by some defining properties. For example, the set of natural numbers can be written as ℕ = {x : x, a natural number} or {x|x, a natural number}.