22. Every Metric Space can be Isometrically Embedded in a Complete Metric Space - I

We offer two proofs that every Metric Space can be Isometrically Embedded in a Complete Metric Space. The first has the virtue of being modeled on the usual proof using Cauchy sequences. It's downside is in its complexity since the members of the Completion are equivalence classes of Cauchy sequences.

22.1 Definitions, Remarks, Observations, and Notation:

1. We will use the notation as short hand for . We will also use the notation for indexed sets of sequences

2. Given a Metric Space and subsets , we define

   and

   In general takes values in

3. Given sequences ₁and it will be convenient to define

   

4. Finally we define

   

   Note that if is finite for any n ,then is finite since

   0

5. Note that, almost by definition, a sequence is a Cauchy sequence if and only if 0.

6. Given a Metric Space we let denote its Set of Cauchy Sequences.

--------------------------------------------------------------------------------------------

22.2 Theorem:

is well defined on the Set ,and satisfies the following metric properties :

1. 0

2. 

3. 

is called a psudo-metric since one can have 0 but

Proof:

To show that is well defined. It suffices to show that

is bounded for some n. Since and are Cauchy,

choose 0 , and n such that for 1 or 2 and n

thus,

2. is immediate. To prove 3. observer that, in general,

hence

------------------------------------------------------------------------

22.3 Definitions, Observations, and Notation:

1. Based on 22.2, we can define an equivalence relation on by setting 0. We use the notation for .

2. One checks that induces a metric on . We use the same notation for this induced metric. On the other hand, we will use to denote the equivalence class of Cauchy sequences containing .

3. Define a Set map by the formula

   , the equivalence class of ,where for all

We will want the following calculation.

22.4 Lemma:

Given a Cauchy sequence ,for each n can can define a sequence where This is called the n tail of . Then

1. is Cauchy

2. .

Proof:

Easy, To Be Turned in (Due Wed. April 20)

----------------------------------------------------------------------

Solution:

For any , so

As well as,

.

Thus

0

---------------------------------------------------------------------------

22.5 Theorem:

is a Complete Metric Space. Moreover, is a dense isometric embedding.

Proof:

To verify that is an isometric embedding note that for any n and

and thus

That the embedding is dense, is the usual argument that a Cauchy sequence is the limit of its terms.

Next, given a Cauchy sequence in we define a sequence in as follows:

• Choose Possibily by replacing by a suitable we can assume that

• Let , the first term of

One next verfies that is Cauchy and

That is Cauchy follows from the observation that, in general

For any n.

In the limit

which goes to 0 as and .

To show that is a similar argument

Also

But

which goes to 0 as