Basic Concepts of Functional Analysis

Abstract

Let X and Y be arbitrary nonempty sets and suppose that some rule is given whereby to each element x ∈ X corresponds a uniquely determined element y ∈ Y which is called the image of x in Y. Consider the set F consisting of all ordered pairs (x,y) where x ∈ X and y is the corresponding element of Y. Such set of pairs is called a function or an operator on X into Y [21]. The set X is called the domain of the function and the set of the elements of Y corresponding to all the elements of X is called the range of the function.