Complete metric space

  • Thread starter Amok
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  • #1
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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?
 

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  • #2
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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 [tex]\mathbb{R}[/tex] is complete: every sequence that should converge converges, but [tex]\mathbb{Q}[/tex] is incomplete, indeed a rational sequence that converges to [tex]\pi[/tex] does not converge in [tex]\mathbb{Q}[/tex].
 
  • #3
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But cam you show that a Hilbert space or a [tex]\mathbb{R}[/tex] space converges? Using the definition of distance, for example? Can you show that every Cauchy sequence in a certain space converges?
 
  • #4
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But cam you show that a Hilbert space or a [tex]\mathbb{R}[/tex] 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 [tex]\mathbb{R}[/tex], but that's because the definition of [tex]\mathbb{R}[/tex] is quite complicated...
 

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