1550251515-Classical_Complex_Analysis__Gonzalez_

92 Chapter 2

The extended complex number system C with the chordal metric as
defined in (1.17-14), (1.17-15), and with x( oo, oo). = 0 is also a metric space.
We shall use the notation ( C, x) for this space. Of course, the same metric

x, as defined in (1.17-14), may also be used in C. The corresponding space

will be denoted (C, x).
In addition to (1.17-16) other metrics can be introduced in C. For
instance, if z =.x + iy,z^1 = x^1 + iy^1 , we may define

p(z,z

1

) =Ix - x

1

I + IY-Y

1

1

This is sometimes called the city-block distance because of its obvious
geometrical interpretation.

4. The set of ordered n-tuples of real numbers x = ( x1, x2, ... , Xn) with
   distance function defined by

[

n ]%

d(x,y) = t;(xk -Yk)^2

is called the Euclidean n-dimensional space, denoted (~n, d). The case
n = 1 gives (~1, d), and the space (~^2 , d) is the same as (C, d) (as a metric
space).
The same set of ordered n-tuples of real numbers with distance

and p ;:::: 1 is called pseudo-Euclidean n-dimensional space, denoted
(~n,dp)·
The same set may also be made into a metric space by introducing the
metric

p(x, y) =max k Jxk -Ykl

5. The set C[a, b] of all continuous real-valued functions defined on the
   interval [a, b] with distance function

d(f,g) = max Jf(t) - g(t)J

a:=;t::;b

is a metric space, denoted simply C[a, b].
The same set with distance function defined by

{

b }

1

/2

p(f,g) = 1 [f(t) - g(t)]^2 dt