Abstract Analysis

Abstract

The operations of elementary algebra are of finite character; that is to say, an algebraic expression depends, explicitly or implicitly, only on a finite number of variables. The step from algebra to analysis is taken when we begin to consider expressions depending on an infinite number of variables; for instance, the limit of a sequence (an) depends on the infinite set of variables a1, a2, …. Such an infinite set of variables is most conveniently thought of as the set of values of a function; thus the sequence (an) is a function of the positive integral variable n, the single term a3, for example, being the value of the function when n is given the particular value 3. Thus the characteristic features of analysis are (i) the systematic use of functions of a variable capable of an infinity of distinct values, and (ii) the consideration of expressions that involve the values of such a function for an infinity of values of the independent variable.