Page 325 - Advanced Linear Algebra

Metric Spaces 309

Example 12.7 The space M B is not separable. Recall that M B is the set of all
(
)
bounded sequences of real numbers or complex numbers with metric
² Á ³ ~ sup( % c & (
%&

To see that this space is not separable, consider the set of all binary sequences
:
:~ ¸²% ³ “ % ~ or for all ¹

This set is in one-to-one correspondence with the set of all subsets of and so
o
. Now, each sequence in is
is uncountable. It has cardinality 2 L €L ³ :
(
certainly bounded and so lies in M B . Moreover, if % £ &  M B , then the two
sequences must differ in at least one position and so ²%Á &³ ~ .
In other words, we have a subset of M B that is uncountable and for which the
:
distance between any two distinct elements is . This implies that the balls in the

uncountable collection ¸)² Á ° ³ “  :¹ are mutually disjoint. Hence, no
countable set can meet every ball, which implies that no countable set can be
dense in M B .
:
Example 12.8 The metric spaces are separable, for M ‚ . The set of all
sequences of the form
~ ² ÁÃÁ Á Áó

for all € , where the 's are rational, is a countable set. Let us show that it is

dense in . Any  % M satisfies
M
B
( % (  B
~
5
Hence, for any € , there exists an such that
B

( % ( 

~5b
Since the rational numbers are dense in , we can find rational numbers for
s

which
( (%c 
5
for all ~ Á ÃÁ5 . Hence, if ~ ² ÁÃÁ Á Áó , then
5

5 B
² % Á ³ ~ % c ( ( b % ( (  b ~
~ ~5b

which shows that there is an element of arbitrarily close to any element of .
:
M

:
Thus, is dense in and so is separable.
M
M