544 Chapter 9 • Sequences and Series of F\mc;tions
Examples 9.1.5 The following are among the many subspaces of F(S, JR):
(a) If Sis any set, B(S) =the set of all bounded^2 real-valued functions on S.
(b) If S is any set of real numbers, we define
(i) C(S) = the set of all real-valued functions that are continuous on S.
(ii) D(S) =the set of all real-valued functions that are differentiable on S.
(iii) Ck(S) =the set of all real-valued functions that have continuous kth
derivative on S.
(iv) C^00 (S) =the set of all real-valued functions that are infinitely differ-
entiable on S.
( c) If S = [a, b] is any compact interval, we define R[a, b] to be the set of all
real-valued functions that are Riemann integrable on [a, b].
(d) If F 1 is any subspace of F(S, JR) and x 0 is any member of S, then the set
of all f E F 1 such that f ( xo) = 0 is a subspace of F1.
Note that all subspaces listed in (b) above are subspaces of B(S), defined in
(a).POINTWISE CONVERGENCE
Our first concern in this chapter is the notion of convergence of a sequence
Un} of functions. We must clarify what we mean by the limit statementn-+oo lim fn = f
when {fn} is a sequence of functions in F(S, JR) and f E F(S, JR). The simplest
notion of convergence is called "pointwise" convergence, which we now define.
Other types of convergence will be defined later.
Definition 9.1.6 (Pointwise Convergence) For a given sequence {fn} of
functions in F(S, JR) and a function f E F(S, JR), we say that fn converges
pointwise to f if\fx ES, fn(x)----> f(x). We sometimes indicate this by writingf n ----> f (pointwise).
We say that a given series I::: fn of functions in F(S, JR) converges point-
wise to a function f E F(S, JR) if its sequence of partial sums converges point-
wise to f. Thus, when we say that a power series "represents" a function f on
an interval we are saying that it converges pointwise to f on that interval.While this type of convergence has a simple definition, it is not the most
useful concept of convergence, as we shall see when we explore some examples.- See Definition 4.2.6.