Page 318 - Advanced Linear Algebra

302 Advanced Linear Algebra

Example 12.1 Any nonempty set 4 is a metric space under the discrete
metric, defined by

% ~ if &
²%Á &³ ~ F
% £ if &
Example 12.2
)
1 The set s is a metric space, under the metric defined for % ~ ²% ÁÃÁ% ³

and & ~ ²& ÁÃÁ& ³ by

²%Á &³ ~ j ²% c & ³ bÄb²% c & ³

This is called the Euclidean metric on s . We note that s is also a metric
space under the metric
(

²%Á &³ ~ % c& bÄb % c& ( ( (
Of course, ² Ás ³ and ² Ás ³ are different metric spaces.
)
2 The set d is a metric space under the unitary metric
²%Á &³ ~ k ( % c& ( bÄb % c& ( (
where % ~ ²% ÁÃÁ% ³ and & ~ ²& Á ÃÁ& ³ are in d .

Example 12.3
)
(
)
1 The set *´ Á µ of all real-valued or complex-valued continuous functions
on ´ Á µ is a metric space, under the metric
² Á ³ ~ sup ( ²%³ c ²%³(
%´ Á µ
We refer to this metric as the sup metric .
)
)
2 The set *´ Á µ of all real-valued or complex-valued continuous functions
²
on ´ Á µ is a metric space, under the metric

(
² ²%³Á ²%³³ ~ ( ²%³ c ²%³ %

Example 12.4 Many important sequence spaces are metric spaces. We will
often use boldface italic letters to denote sequences, as in % ~²% ³ and

& ~²& ³.

)
1 The set M B of all bounded sequences of real numbers is a metric space
s
under the metric defined by
%&
² Á ³ ~ sup( % c & (

The set M B of all bounded complex sequences, with the same metric, is also
d
a metric space. As is customary, we will usually denote both of these spaces
by M B .