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Banach space vs. closed space

  1. Jan 19, 2008 #1
    Hi to all
    What exactly is the difference between Banach(=complete, as far as I understand) (sub)space and closed (sub)space. Is there a normed vector space that is complete but not closed or normed vectore space that is closed but not complete?
    Thanks in advance for explanation and/or examples.
     
  2. jcsd
  3. Jan 19, 2008 #2

    EnumaElish

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    Is this homework?
     
  4. Jan 19, 2008 #3
    No, it is not. While studying some proofs I relized that I know two different definitions for the two different things but I can't really put my finger on the differences, if there are any.
    But I do not see the purpose of your question, except that you would 'educate' me that I posted in a wrong forum in case this were homework.
     
  5. Jan 19, 2008 #4

    mathwonk

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    complete is an absolute term, i.e. either a space is complete or it isn't.

    closed is a relative term, i.e. a subspace is closed in some other space, or not.

    but the same space can be closed in one space and non closed in another.

    i.e. closed is a property of a pair of spaces.

    this is obviously not hw as it is too basic. no professor would dream of asking this since they would just assume it is understood.
     
  6. Jan 19, 2008 #5

    quasar987

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    Given a normed space X, a subspace E of X is

    1) Banach(=complete) if every Cauchy sequence in E converge to a point of E.

    2) Closed if every sequence in E that converge in X, converge to a point of E.

    If you want to make sure you understand the distinction and relation between the two, prove these two elementary observations: "If X is Banach and E is closed, then E is Banach" and this: "If E is Banach, then is it closed."
     
  7. Jan 20, 2008 #6
    Thanks to you both, I think I do understand the difference now. As for the observations, the proofs could be as follows:

    1) Since X is Banach, a given Cauchy sequence in E (which must then also be in X) converges to a point in X and since E is closed, every sequence from E that converges in X has a limit in E - and so has our Cauchy sequence. Summary: any given Cauchy sequence in E has limit in E which is the definition of completness.

    2) Since E is Banach, every Cauchy seq. from E has limit in E. Also every convergent (with limit in X, generally) sequence in E must be Cauchy sequence -> these two together imply that every convergent sequence from E must have limit in E which is what I want to prove.

    Correct me if I am wrong, H.
     
  8. Jan 21, 2008 #7

    quasar987

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    Flawless. :)
     
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