Page 317 - Advanced Linear Algebra

Chapter 12

Metric Spaces

The Definition

In Chapter 9, we studied the basic properties of real and complex inner product
spaces. Much of what we did does not depend on whether the space in question
is finite-dimensional or infinite-dimensional. However, as we discussed in
Chapter 9, the presence of an inner product and hence a metric, on a vector
space, raises a host of new issues related to convergence. In this chapter, we
discuss briefly the concept of a metric space. This will enable us to study the
convergence properties of real and complex inner product spaces.

A metric space is not an algebraic structure. Rather it is designed to model the
abstract properties of distance.

Definition A metric space is a pair ²4Á ³ , where 4 is a nonempty set and
¢ 4 d 4 ¦ s is a real-valued function, called a metric on 4, with the
%
&
following properties. The expression ²%Á &³ is read “the distance from to .”
1 )(Positive definiteness ) For all %Á &  4 ,
²%Á &³ ‚
and ²%Á &³ ~ if and only if % ~ & .
2 )(Symmetry ) For all %Á &  4 ,

²%Á &³ ~ ²&Á %³
3 )(Triangle inequality ) For all %Á &Á '  4 ,

²%Á &³  ²%Á '³ b ²'Á &³
As is customary, when there is no cause for confusion, we simply say “let 4 be
a metric space.”