PRELIMINARIES

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.

1.1 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 [member of] 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}. Next we discuss set inclusion. If every element of a set A is an element of the set B, A is said to be a subset of the set B or B is said to be a superset of A, and this is denoted by A [??] B or B [??] A. Two sets A and B are said to be equal if every element of A is an element of B and every element of B is an element of A – in other words if A [??] B and B [??] A. If A is equal to B, then we write A = B. A set is generally completely determined by its elements, but there may be a set that has no element in it. Such a set is called an empty (or void or null) set and the empty set is denoted by Φ (Phi). Φ [subset] A; in other words, the null set is included in any set A – this fact is vacuously satisfied. Furthermore, if A is a subset of B, [A ≠ Φ and A ≠ B then A is said to be a proper subset of B (or B is said to properly contain A). The fact that A is a proper subset of B is expressed as A [subset] B. Let A be a set. Then the set of all subsets of A is called the power set of A and is denoted by P(A). If A has three elements like letters p, q and r, then the set of all subsets of A has 8(= 23) elements. It may be noted that the null set is also a subset of A. A set is called a finite set if it is empty or it has n elements for some positive integer n; otherwise it is said to be infinite. It is clear that the empty set and the set A are members of P (A). A set A is called denumerable or enumerable if it is in one-to-one correspondence with the set of natural numbers. A set is called countable if it is either finite or denumerable. A set that is not countable is called uncountable.

We now state without proof a few results which might be used in subsequent chapters:

(i) An infinite set is equivalent to a subset of itself.

(ii) A subset of a countable set is a countable set.

The following are examples of countable sets: a) the set J of all integers, b) the set Q of all rational numbers, c) the set P of all polynomials with rational coefficients, d) the set all straight lines in a plane each of which passes through (at least) two different points with rational coordinates and e) the set of all rational points in Rn.

Examples of uncountable sets are as follows: (i) an open interval [a, b], a closed interval [a, b] where a ≠ b, (ii) the set of all irrational numbers. (iii) the set of all real numbers. (iv) the family of all subsets of a denumerable set.

1.1.1 Cardinal numbers

Let all the sets be divided into two families such that two sets fall into one family if and only if they are equivalent. This is possible because the relation ~ between the sets is an equivalence relation. To every such family of sets, we assign some arbitrary symbol and call it the cardinal number of each set of the given family. If the cardinal number of a set A is α, A = α or card A = α The cardinal number of the empty set is defined to be 0 (zero). We designate the number of elements of a nonempty finite set as the cardinal number of the finite set. We assign N0 to the class of all denumerable sets and as such N0 is the cardinal number of a denumerable set. c, the first letter of the word 'continuum' stands for the cardinal number of the set [0, 1].

1.1.2 The algebra of sets

In the following section we discuss some operations that can be performed on sets. By universal set we mean a set that contains all the sets under reference. The universal set is denoted by U. For example, while discussing the set of real numbers we take R as the universal set. Once again for sets of complex numbers the universal set is the set C of complex numbers. Given two sets A and B, the union of A and B is denoted by A [union] B and stands for a set whose every element is an element of either A or B (including elements of both A and B). A [union] B is also called the sum of A and B and is written as A + B. The intersection of two sets A and B is denoted by A [intersection] B, and is a set, the elements of which are the elements common to both A and B. The intersection of two sets A and B is also called the product of A and B and is denoted by A x B. The difference of two sets A and B is denoted by A-B and is defined by the set of elements in A which are not elements of B. Two sets A and B are said to...