What is a Complete Metric Space in Mathematics?

[Amok]

Can someone help me understand the notion of complete metric space? I've read the definition (the one involving metrics that go to 0), but I can't really picture what it is. Does anyone have any examples that could help me understand this?

 

[micromass]

I'm not sure if there even is something you can picture. Completeness is simply a quite technical condition that has a lot of benifits. Intuitively, one can say that a space is complete if every sequence that should converge, also converges.
What is a sequence that should converge? Well a sequence who's terms lie closer and closer together. For example, the sequence (1/n) should converge, because the terms are closer and closer. But (n) does not converge, because the terms both have distance 1 from each other.

The space $$\mathbb{R}$$ is complete: every sequence that should converge converges, but $$\mathbb{Q}$$ is incomplete, indeed a rational sequence that converges to $$\pi$$ does not converge in $$\mathbb{Q}$$.

 

[Amok]

But cam you show that a Hilbert space or a $$\mathbb{R}$$ space converges? Using the definition of distance, for example? Can you show that every Cauchy sequence in a certain space converges?

 

[micromass]

But cam you show that a Hilbert space or a $$\mathbb{R}$$ space converges? Using the definition of distance, for example?

What do you mean with "a Hilbert spaces converges"?

Can you show that every Cauchy sequence in a certain space converges?

Yes, one can show that for a lot of spaces, so it's certainly not an impossible condition to check. The only space for which it is really hard to check is for $$\mathbb{R}$$, but that's because the definition of $$\mathbb{R}$$ is quite complicated...