Page 327 - Advanced Linear Algebra
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Metric Spaces 311
such that for all ,
)²% Á ³ )² ²% ³Á ³ 5 \
4
Thus, for all ,
\
) % Á 4 7 5 6 )² ²% ³Á ³
and so we may construct a sequence ²% ³ by choosing each term % with the
property that
7
% ) % Á 6 , but ²% ³ ¤ )² ²% ³Á ³
Hence, ²% ³ ¦ % , but ²% ³ does not converge to ²% ³ . This contradiction
implies that must be continuous at .
%
The next theorem says that the distance function is a continuous function in both
variables.
Theorem 12.5 Let ²4Á ³ be a metric space. If ²% ³ ¦ % and ²& ³ ¦ & , then
²% Á & ³ ¦ ²%Á &³.
Proof. We leave it as an exercise to show that
( ( ²% Á & ³ c ²%Á &³ ²% Á %³ b ²& Á &³
But the right side tends to as ¦B and so ²% Á & ³ ¦ ²%Á &³ .
Completeness
The reader who has studied analysis will recognize the following definitions.
Definition A sequence ²% ³ in a metric space 4 is a Cauchy sequence if for
any , there exists an 5 for which
Á 5 ¬ ²% Á % ³
We leave it to the reader to show that any convergent sequence is a Cauchy
sequence. When the converse holds, the space is said to be complete .
Definition Let 4 be a metric space.
1 4 ) is said to be complete if every Cauchy sequence in 4 converges in 4 .
2 A subspace of 4 is complete if it is complete as a metric space. Thus, :
)
:
:
is complete if every Cauchy sequence ² ³ in converges to an element in
:.
Before considering examples, we prove a very useful result about completeness
of subspaces.
