Banach Spaces vs. Closed Spaces: What's the Difference?

[hooker27]

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.

 

[EnumaElish]

Is this homework?

 

[hooker27]

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.

 

[mathwonk]

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 homework as it is too basic. no professor would dream of asking this since they would just assume it is understood.

 

[quasar987]

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."

 

[hooker27]

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.

 

[quasar987]

Flawless. :)