Page 329 - Advanced Linear Algebra
P. 329
Metric Spaces 313
(
Example 12.11 The metric space ²*´ Á µÁ ³ of all real-valued or complex-
)
valued continuous functions on ´ Á µ , with metric
² Á ³ ~ sup ( ²%³ c ²%³(
%´ Á µ
is complete. To see this, we first observe that the limit with respect to is the
uniform limit on ´ Á µ , that is ² Á ³ ¦ if and only if for any , there is
an 5 for which
5 ¬ ²%³ c ²%³ for all %´ Á µ
(
(
Now let ² ³ be a Cauchy sequence in ²*´ Á µÁ ³ . Thus, for any , there is
an for which
5
Á 5 ¬ ²%³ c ²%³ for all % ´ Á µ ( 12.1)
(
(
This implies that, for each %´ Á µ , the sequence ² ²%³³ is a Cauchy sequence
)
of real or complex numbers and so it converges. We can therefore define a
(
function on ´ Á µ by
²%³ ~ lim ²%³
¦B
(
)
Letting ¦B in 12.1 , we get
5 ¬ ²%³ c ²%³ for all %´ Á µ
(
(
Thus, ²%³ converges to ²%³ uniformly. It is well known that the uniform
limit of continuous functions is continuous and so ²%³ *´ Á µ . Thus,
² ²%³³ ¦ ²%³ *´ Á µ and so ²*´ Á µÁ ³ is complete.
(
Example 12.12 The metric space ²*´ Á µÁ ³ of all real-valued or complex-
)
valued continuous functions on ´ Á µ , with metric
² ²%³Á ²%³³ ~ ( ²%³ c ²%³ %
(
is not complete. For convenience, we take ´ Á µ ~ ´ Á µ and leave the general
case for the reader. Consider the sequence of functions ²%³ whose graphs are
(
shown in Figure 12.3. The definition of ²%³ should be clear from the graph.³
