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

In summary, the conversation discusses the difference between a Banach (complete) subspace and a closed subspace in a normed vector space. The key distinction is that completeness is an absolute term, while closedness is relative to another space. It is possible for a subspace to be complete but not closed or vice versa. The conversation also includes a discussion on the definitions and properties of these two concepts, with two elementary observations being proposed for further understanding.
  • #1
hooker27
16
0
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.
 
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  • #2
Is this homework?
 
  • #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.
 
  • #4
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.
 
  • #5
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."
 
  • #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.
 
  • #7
Flawless. :)
 

1. What is the difference between a Banach space and a closed space?

A Banach space is a complete normed vector space, meaning that every Cauchy sequence in the space converges to a point within the space. A closed space, on the other hand, is a vector space that contains all of its limit points. Therefore, all Banach spaces are closed spaces, but not all closed spaces are Banach spaces.

2. How are Banach spaces and closed spaces used in mathematics?

Banach spaces and closed spaces are used in many areas of mathematics, including functional analysis, differential equations, and harmonic analysis. They provide a framework for studying infinite-dimensional spaces and are essential in understanding the behavior of functions and operators.

3. Can a Banach space be non-closed?

No, a Banach space must be closed. This is because a Banach space is defined as a complete normed vector space, meaning that it contains all of its limit points. If a Banach space were not closed, it would not contain all of its limit points and would therefore not be complete.

4. Are there any examples of Banach spaces that are not closed?

No, there are no examples of Banach spaces that are not closed. This is because, as mentioned before, a Banach space must be closed in order to be considered complete. Any example that may seem like a Banach space but is not closed would not meet the definition of a Banach space.

5. How are Banach spaces and closed spaces related to compact spaces?

Banach spaces and closed spaces are not directly related to compact spaces. Compact spaces are topological spaces that have the property that every open cover has a finite subcover. However, it is possible for a Banach space or a closed space to be compact. In fact, in finite-dimensional vector spaces, the notions of Banach spaces, closed spaces, and compact spaces all coincide.

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