Page 334 - Advanced Linear Algebra

318 Advanced Linear Algebra

and so
Z
Z
Z
² Á ³  ² Á ³ b ² Á ³
Z
Z
Embedding ²4Á ³ in ²4 Á ³
For each %  4 , consider the constant Cauchy sequence ´%µ , where ´%µ² ³ ~ %

for all . The map ¢ 4 ¦ 4 Z defined by
%~´%µ
is an isometry, since

Z
² %Á &³ ~ ²´%µÁ ´&µ³ ~ lim ²´%µ² ³Á ´&µ² ³³ ~ ²%Á &³
Z

¦B
Moreover, 4 is dense in 4 Z . This follows from the fact that we can
approximate any Cauchy sequence in 4 by a constant sequence. In particular,

let  4 Z . Since  is a Cauchy sequence, for any € , there exists an 5
such that
Á ‚ 5 ¬ ² ² ³Á ² ³³ 
Now, for the constant sequence ´ ²5³µ we have

Z
´ ²5³µÁ ~ lim ² ²5³Á ² ³³ 
4
5
¦B
and so 4 is dense in 4 Z .
Z
Z
²4 Á ³ Is Complete
Suppose that
Á Á Á Ã
3

is a Cauchy sequence in 4 Z . We wish to find a Cauchy sequence in 4 for

which
² Á ³ ~ lim ² ² ³Á ² ³³ ¦ as ¦ B
Z

¦B
Since  4 Z and since 4 is dense in 4 Z , there is a constant sequence

´ µ ~ ² Á Áó

for which

Z
² Á ´ µ³