Chapter 9
Sequences and Series of
Functions
The chief concern of this chapter is uniform convergence and
its consequences. By using the notion of distance between
functions we set the stage for the study of function spaces
in more advanced courses. In Section 9.4 we bring the course
to a culmination in two famous theorems of Weierstrass: on
the existence of continuous, nowhere differentiable functions,
and on polynomial approximation of continuous functions.In many areas of advanced analysis, significant power is gained by shifting our
attention from sequences, series, and sets of numbers to sequences, series and
"spaces" of functions. We begin this shift of attention here, by considering
families of functions in Section 9.1. The meaning of convergence of a sequence
of functions will be considered in Section 9.1, but a more satisfactory type of
convergence will be defined in Section 9.2.
9.1 Families of Functions and Pointwise
Convergence
Definition 9.1.1 Let S denote an arbitrary set. Any function f : S -+JR is
called a real-valued function on S. We shall consider the set of all such
functions,
F (S, JR)= {all functions f: S -+JR}.
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