Page 336 - Advanced Linear Algebra
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320 Advanced Linear Algebra
Figure 12.4
Our goal is to show that can be extended to a bijective isometry from 4 Z to
ZZ
4 .
Let % 4 Z . Then there is a sequence ² ³ in 4 for which ² ³ ¦ % . Since
² ³ is a Cauchy sequence in 4 ² ² ³³ is a Cauchy sequence in , 4 4 ZZ
and since 4 ZZ is complete, we have ² ² ³³ ¦ & for some & 4 ZZ . Let us
define ²%³ ~ & .
To see that is well-defined, suppose that ² ³ ¦ % and ² ³ ¦ % , where both
sequences lie in 4 . Then
Z
² ² ³Á ² ³³ ~ ² Á ³ ¦ as ¦ B
ZZ
and so ²² ³³ and ²² ³³ converge to the same element of 4 ZZ , which implies
that does not depend on the choice of sequence in 4 converging to .
²%³
%
Thus, is well-defined. Moreover, if 4 , then the constant sequence ´ µ
converges to and so ² ³ ~ lim ² ³ ~ ² ³ , which shows that is an
extension of .
To see that is an isometry, suppose that ² ³ ¦ % and ² ³ ¦ & . Then
ZZ
² ² ³³ ¦ ²%³ and ² ² ³³ ¦ ²&³ and since is continuous, we have
ZZ
ZZ
Z
Z
² ²%³Á ²&³³ ~ lim ² ² ³Á ² ³³ ~ lim ² Á ³ ~ ²%Á &³
¦B ¦B
Thus, we need only show that is surjective. Note first that
² ³ im
4~ im ² ³. Thus, if im ² ³ is closed, we can deduce from the fact
that is dense in 4 ZZ that im 4 ZZ . So, suppose that ²³ ~ 4 ²²% ³³ is a
sequence in im²³ and ²²% ³³ ¦ ' . Then ²²% ³³ is a Cauchy sequence and
therefore so is ²% ³ . Thus, ²% ³ ¦ % 4 Z . But is continuous and so
²²% ³³ ¦ ²%³, which implies that ²%³ ~ ' and so ' im ²³. Hence, is
Z
surjective and 4 4 Z Z .
