1549901369-Elements_of_Real_Analysis__Denlinger_

9.5 *A Glimpse Beyond the Horizon 579

such as C[a, b]. But there are other norms commonly in use on C[a, b] and its

subspaces; for example,

111111 =1: lfl, and

111112=v1: 111

2

·

Each of these norms on C[a, b] is useful for certain purposes, such as in

Fourier analysis or approximation theory. You may wish to check for yourself

that these are indeed norms on [a, b]. It is fairly easy to understand why 11·11 1

might be useful as a norm, when you remember that the 111111 is intended to

indicate the distance of f from the 0 function, and that II! -gll is intended

to represent the distance between f and g. The area between their graphs is a

reasonable measure of the distance between two functions. See Figure 9.16.

y

11/11 1 = total area

x

llf-gll 1 =total area between curves.

Figure 9.16

The importance of the second norm, 11·1'2, can perhaps be best understood
when it is seen as more closely resembling the familiar "Euclidean" norm on
]Rn,

ll(x1,x2, · .. ,xn)ll = \/~ x;.

This norm, 11 · 112 , is commonly used in approximation theory and Fourier anal-
ysis, although it is not the only one.

Many of the concepts learned in this course generalize to normed vector
spaces. For example, a sequence { Vn} in a normed vector space V is said to
converge to an element v E V if
'tic> 0, :3 no E .N 3 n 2: no=} llvn - vii < c (i.e., llvn -vii -t 0).
A sequence { Vn} in a normed vector space V is said to be a Cauchy sequence
if

'tic> 0, :3 no E .N 3 m, n 2: no==} llvm - vnll < c.
Thus, a sequence Un} of functions in the space C[a, b] converges to a
function f by this definition iff fn __, f uniformly on [a,b]. Theorems 9.2.7 and