1550251515-Classical_Complex_Analysis__Gonzalez_

Topology of Plane Sets of Points 93

is also a metric space, denoted 02 [a, b]. It is called the space of continuous

functions with quadratic metric.

Let (S,d) be any metric space and let M be a nonempty subset of S. If

d M is the restriction of the function d to M, i.e., if d M ( x, y) = d( x, y) for

(x, y) E M x M, it is a trivial matter to verify that dM is a metric in M.

Thus (M, dM) is a metric space, and it is referred to as a subspace of S.

The metric dM is called the relative metric induced on M by d.

EXERCISES 2.3

1. Show that the following metrics satisfy the axioms of Definition 2.7.
   (a) d(x,y) = Ix - YI on IB.^1
   (b) d(z, z^1 ) = lz - z^1 I on C
   (c) x(z,z^1 ) on C*
   (d) p(z,z^1 ) = Ix - x^1 I + IY - y^1 I on C
   (e) p(x,y) = maxlxk - Ykl on IB.n

(f) d(f,g) = maxlf(t)-g(t)I, a:::; t :::;· b, on O[a,b]

2. Prove that if d( x, y) is a metric in a set S, then
   d(x,y)
   p(x,y) = 1 + d(x,y)

i:;i also metric in the same set.

3. If, as in problem 2, d(x, y) is a metric in S, prove that

p(x, y) = min[l, d(x, y)]

is also a metric in S.

4. Show that the four axioms of Definition 2. 7 can be reduced to the
   following two axioms (A. Lindenbaum, 1926):
   (a) d( x, y) = 0 iff x = y
   (b) d(x,z):::; d(x,y) + d(z,y) for arbitrary x,y,z ES


5. For any complex number z, let !!zl! = p(z, 0) = lxl + IYI· Prove the
   following.
   (a) l!z1 + z21! :::; !lz1!1 + !lz2ll
   (b) l!z1z2!1 :::; l!z1!1 · l!z21!
   (c) !z!2 :::; l!zl!2 :::; 2!z!2

(d) 1/!lz!I :::; 111/zll :::; 2/!!zl! (z -f 0)

2.6 NEIGHBORHOODS. OPEN AND CLOSED SETS

Let (S,d) be any metric space, and let a be any given point in S.

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