Sequences and Series of Functions

Abstract

In Chapter 2, we studied sequences and series of (constant) real numbers. In most problems, however, it is desirable to approximate functions by more elementary ones that are easier to investigate. We have already done this on a few occasions. For example, in Chapter 4, we looked at the uniform approximation of continuous functions by step, piecewise linear, and polynomial functions. Also, in Chapter 7, we proved that each bounded continuous function on a closed bounded interval is a uniform limit of regulated functions. Now all these approximations involve estimates on the distance between the given continuous function and the elementary functions that approximate it. This in turn suggests the introduction of sequences (and hence also series) whose terms are functions defined, in most cases, on the same interval. Throughout this chapter, we shall assume that I, possibly with subscript, is an interval of ℝ. Although we are studying Real Analysis here, we should at least introduce the field ℂ of complex numbers and even use it in some definitions if this clarifies the concepts. Our presentation will be brief and most of the proofs are left as simple exercises for the reader.