Number Theory: An Introduction to Mathematics

4 Metric Spaces 29

(iii) LetE=C(I)be the set of all continuous functionsf:I→R,where

I=[a,b]={x∈R:a≤x≤b}

is an interval ofR, and define d(g,h)=|g−h|,where

|f|= sup

a≤x≤b

|f(x)|.

(A well-known property of continuous functions ensures that f is bounded onI.)
Alternatively, one can replace thenorm|f|by either

|f| 1 =

∫b

a

|f(x)|dx

or

|f| 2 =

(∫b

a

|f(x)|^2 dx

) 1 / 2

.

(iv) LetE=C(R)be the set of all continuous functionsf:R→Rand define

d(g,h)=

∑

N≥ 1

dN(g,h)/ 2 N[1+dN(g,h)],

where dN(g,h)=sup|x|≤N|g(x)−h(x)|. The triangle inequality(D3)follows from
the inequality

|α+β|/[1+|α+β|]≤|α|/[1+|α|]+|β|/[1+|β|]

for arbitrary real numbersα,β.
The metric here has the property that d(fn,f)→0 if and only iffn(x)→ f(x)
uniformly on every bounded subinterval ofR. It may be noted that, even thoughEis
a vector space, the metric is not derived from a norm since, ifλ∈R, one may have
d(λg,λh)=|λ|d(g,h).

(v) LetEbe the set of all measurable functionsf:I →R,whereI=[a,b]isan
interval ofR, and define

d(g,h)=

∫b

a

|g(x)−h(x)|( 1 +|g(x)−h(x)|)−^1 dx.

In order to obtain(D1), we identify functions which take the same value at all points
ofI, except for a set of measure zero.
Convergence with respect to this metric coincides withconvergence in measure,
which plays a role in the theory of probability.

(vi) LetE=F∞ 2 be the set of all infinite sequencesa=(α 1 ,α 2 ,...),whereαj=0or
1foreveryj, and define d(a,a)=0, d(a,b)= 2 −kifa=b,whereb=(β 1 ,β 2 ,...)
andkis the least positive integer such thatαk=βk.