1549901369-Elements_of_Real_Analysis__Denlinger_

9.2 Uniform Convergence 549

(See Figure 9.6.) Thus, l\fll is the distance between f and the 0 function.

y

y

y=g(x)

x

x y=f(x)

Figure 9.5

Figure 9.6

Thus, for a given E: > 0, a function g E B(S) is within a distance E: of a

function f E B(S) if its graph lies within a "band" E: units above and below

the graph off. See Figure 9.7.

y

Figure 9.7

The "sup norm" has several important algebraic properties which make it
resemble the absolute value, and that form the basis of its usefulness.

Theorem 9.2.2 Given any f,g E B(S), and any real number r,

(a) llfll ~ 0, and llfll = 0 if and only if f = O;

(b) Iii+ gll :::; l\fll + 11911 (triangle inequality);

(c) llr fll = lrl llfll;

(d) llfgll:::; l\fll 11911-

Proof. Exercise l. • (See also Exercise 2.)

We are finally ready to state the definition of "uniform convergence" of a
sequence of functions. Although this definition seems complicated, we shall find
that by using the notion of "norm" we can make it more understandable.