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)
[edit] Strange Example
In the text there is the example "The topological vector space R^ω of sequences of real numbers which have finitely many nonzero terms (the topology of this space can alternatively be defined as the limit topology of the Rn or as the coproduct of infinitely many copies of R) is not complete (even though its underlying field is). The completion of this space is the product ΠR of infinitely many copies of R. If instead we endow the space with the lp norm, its completion is the space lp(N)."
This contains many false statements, like
1.) The beginning "The topological vector space R^ω of sequences of real numbers which have finitely many nonzero terms" is already a bit strange, since so far no topology is given and therefore R^ω is just a vector space and not yet a topological one. There are many totally different topologies on this vector space.
2.) "the topology of this space can alternatively be defined as the limit topology of the Rn or as the coproduct of infinitely many copies of R" Yes, it is possible to endow this vector space with this particular topology, but then is not metrizable (and therefore not at all a good example for a "Complete Metric Space"). By the way, the space R^ω endowed with this topology is complete but not with respect to a metric (since there is none defining this topology) but with respect to the uniform strucutre as a topological vector space.
3.) The completion of this coproduct in number (2.) is NOT the product ΠR, since the coproduct is already complete as mentioned above. It is true that the product can be the completion of R^ω, but with another topology (which that in fact will be again metrizable)
4.) I admit that the last sentence is correct. "If instead we endow the space with the lp norm, its completion is the space lp(N)."
I would suggest that we remove this example or at least restrict it to the case where we put an lp-norm on it. --130.83.2.27 08:09, 7 August 2007 (UTC)
- I cut out the example per your item 1 (I am not competent enough to judge items 2 and 3. :) There are many good examples in the text that we can get by without this very technical example, especially that it is not well-written and possibly incorrect. Oleg Alexandrov (talk) 19:43, 7 August 2007 (UTC)
