Isometric embedding from metric space to completion

[radou]

Well, this is actually pretty easy, and we have already proved it.

As stated, a Cauchy sequence in Y corresponds to a Cauchy sequence xn in X. By (c), h(xn) converges to [(x1, x2, ...)] in Y.

 

[micromass]

Have we already proved that? Well, then it's pretty easy indeed!

 

[radou]

Have we already proved that? Well, then it's pretty easy indeed!

Yes, it's basically in post #10, unless I'm mistaken.

 

[micromass]

Ah yes, it is basically the same thing indeed! That completes your completion exercise

 

[radou]

Ah yes, it is basically the same thing indeed! That completes your completion exercise

Yes... So the purpose of this exercise was to show another way to imbed a metric space into a complete metric space, through a relation defined for Cauchy sequences of the original space.

The other way to imbed some metric space was (Theorem 43.7., and actually I dislike this theorem) through the set of all bounded functions from that space into R...i.e. there is an imbedding of (X, d) into the set of all bounded functions from X to R in the uniform metric.

 

[micromass]

Yes, now you've shown that every metric space has a completion. The usual way to prove this is by exercise 9. The reason most textbooks prefer exercise 9 is because it can be easily generalized and because the completion is very easy to describe.

One can in fact also show that the completion is unique. This is actually a consequence of exercise 2.

Also note that $$\mathbb{Q}$$ is an incomplete metric space. It's completion is of course $$\mathbb{R}$$. And exercise 9 now gives a very easy idea of how to construct $$\mathbb{R}$$! Just take all Cauchy sequences of rational numbers...

 

[radou]

Also note that $$\mathbb{Q}$$ is an incomplete metric space. It's completion is of course $$\mathbb{R}$$. And exercise 9 now gives a very easy idea of how to construct $$\mathbb{R}$$! Just take all Cauchy sequences of rational numbers...

Wow, I never thought of it that way! Thanks!

Btw, basically, this doesn't strictly have much to do with topology, right? i.e. it's more about metric spaces...

 

[micromass]

Yes, this is more analysis than topology. In fact, entire chapter 7 seems to be more about metric spaces than topology.

If you're going to study functional analysis, then you're going to see much of chapter 7 again. Specifically, the completion is very important in functional analysis! But you're correct, it's not really topology...