Page 332 - Advanced Linear Algebra

316 Advanced Linear Algebra

for all %Á &  4 . If ¢ 4 ¦ 4 Z is a bijective isometry from 4 to 4 Z , we say
that 4 and 4 Z are isometric and write 4 š 4 Z .

Z
Z
Theorem 12.7 Let ¢ ²4Á ³ ¦ ²4 Á ³ be an isometry. Then
)
1 is injective
)
2 is continuous
3 ) c ¢ ² 4 ³ ¦ 4 is also an isometry and hence also continuous.
Proof. To prove 1 , we observe that
)
Z
²%³ ~ ²&³ ¯ ² ²%³Á ²&³³ ~ ¯ ²%Á &³ ~ ¯ % ~ &
)
To prove 2 , let ²% ³ ¦ % in 4 . Then

Z
² ²% ³Á ²%³³ ~ ²% Á %³ ¦ as ¦ B

and so ² ²% ³³ ¦ ²%³ , which proves that is continuous. Finally, we have

Z
² c ² ²%³³Á c ² ²&³³ ~ ²%Á &³ ~ ² ²%³Á ²&³³
and so c ¢ ² 4 ³ ¦ 4 is an isometry.
The Completion of a Metric Space
While not all metric spaces are complete, any metric space can be embedded in
a complete metric space. To be more specific, we have the following important
theorem.

Theorem 12.8 Let ²4Á ³ be any metric space. Then there is a complete metric
Z
Z

space ²4 Á ³ and an isometry ¢ 4¦ 4‹ 4 Z for which 4 is dense in

4 Z ² 4. The metric space Z Á Z ³ is called a of ² 4 Á completion ³ . Moreover,
²4 Á ³ is unique, up to bijective isometry.
Z
Z
Proof. The proof is a bit lengthy, so we divide it into various parts. We can
simplify the notation considerably by thinking of sequences ²% ³ in 4 as

functions ¢ o ¦ 4 , where ² ³ ~ % .
Cauchy Sequences in 4
The basic idea is to let the elements of 4 Z be equivalence classes of Cauchy
sequences in 4 . So let CS ² 4 ³ denote the set of all Cauchy sequences in 4 . If
Á  CS ²4³, then, intuitively speaking, the terms ² ³ get closer together as
¦B and so do the terms ² ³. Therefore, it seems reasonable that
² ² ³Á ² ³³ should approach a finite limit as ¦ B. Indeed, since
( ( ² ² ³Á ² ³³ c ² ² ³Á ² ³³  ² ² ³Á ² ³³ b ² ² ³Á ² ³³ ¦

as Á ¦ B it follows that ² ² ³Á ² ³³ is a Cauchy sequence of real
numbers, which implies that