Talk:Complete metric space
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I removed this link:
- Cech completeness, absolute G_delta spaces
from the discussion of topologically complete spaces, since it's malformed and didn't fit into the sentence. If anybody knows what it's about, please put it back in as a legible sentence and with a link that might be the title of an article (perhaps it should be two).
I also moved the definition of Cauchy net to its own page.
-- Toby 23:40 Feb 20, 2003 (UTC)
What's all this about inner product spaces? There's nothing in Inner product space about completeness -- except completeness in a metric! If anything's wrong, then it's that this article should have something to say about how the ideas are applied to certain types of metric spaces -- such as inner product spaces. -- Toby 00:50 Feb 21, 2003 (UTC)
OK ... I think I was hasty and actually had in mind the idea of "complete" orthonormal set in an inner product space. Such an "orthonormal basis" is not always a "basis" in the sense of "Hamel basis", i.e., not every vector in such a space is a finite linear combination of basis vectors. And in infinite linear combinations, there are various different notions of convergence, and the one that matters in this case is the one defined by the metric. Michael Hardy 01:05 Feb 21, 2003 (UTC)
Yes, and there's a link to Completeness from Hermitian that's about this. So we should add it to Completeness (I just did). But this is complete orthonormal set, not complete space. -- Toby 03:46 Feb 21, 2003 (UTC)
[edit] complete spaces vs. closed set
from this article i don't really understand what is the difference between a closed set and a complete space. maybe someone can give an example of a closed space which ins't complete? —The preceding unsigned comment was added by 217.132.104.254 (talk) 09:38, 8 February 2007 (UTC).
Spaces aren't closed or open; subspaces are/aren't. Consider the rational numbers, Q, thought of as a topological space with the topology induced from the usual Borel topology on the real line R. Q is a closed subset of itself, since the empty set is an open subset of any topological space. However, Q is not complete; consider the sequence of successive decimal approximations to the square root of 2, which is Cauchy but not convergent. Sullivan.t.j 14:14, 8 February 2007 (UTC)
[edit] Other formulations of the completeness axiom
Is there a reason this specific formulation of the Completeness axiom was chosen over, say, the Supremum formulation, or that of Monotonic Convergence? Should these not at least be mentioned with a discussion of the benefits / downfalls of different formulations? I ask this specifically as the Supremum (or Infimum) formulation is often easier for the non-mathematician to understand. In a sense the most complex formulation is that represented here, and the question is whether this is where Wikipedia articles should be aimed.
-Jdr [13/3/2007 - 10:06pm GMT]
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- "Supremum" makes sense when there is an ordering, but in metric spaces generally there is no supremum. Which "monotic convergence" formultation are you talking about? Does it work in metric spaces generally? "Monotonic" seems to connote an ordering. Michael Hardy 22:54, 13 March 2007 (UTC)
