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

Click For Summary
SUMMARY

The discussion clarifies the distinction between Banach spaces and closed spaces within the context of normed vector spaces. A Banach space is defined as a complete space where every Cauchy sequence converges to a point within the space. In contrast, a closed subspace is characterized by the property that every sequence converging in the larger space converges to a point within the subspace. The conversation emphasizes that while a Banach space is always complete, a closed space may not be complete, and vice versa, highlighting the nuanced relationship between these two concepts.

PREREQUISITES
  • Understanding of normed vector spaces
  • Familiarity with Cauchy sequences
  • Knowledge of convergence in metric spaces
  • Basic principles of functional analysis
NEXT STEPS
  • Study the properties of Cauchy sequences in Banach spaces
  • Explore examples of closed subspaces in various normed spaces
  • Learn about the implications of completeness in functional analysis
  • Investigate theorems relating Banach spaces and closed subspaces
USEFUL FOR

Mathematicians, students of functional analysis, and anyone interested in the foundational concepts of Banach and closed spaces in normed vector spaces.

hooker27
Messages
15
Reaction score
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.
 
Physics news on Phys.org
Is this homework?
 
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.
 
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.
 
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."
 
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.
 
Flawless. :)
 

Similar threads

High School What Are the Philosophical Implications of the Banach-Tarski Paradox?
  • · Replies 26 ·
Replies
26
Views
4K
Help Needed Proving Implication for Linear Functional on Banach Space
  • · Replies 2 ·
Replies
2
Views
1K
High School How Many Straight Lines to Connect an N by M Array of Points in a Closed Loop?
  • · Replies 25 ·
Replies
25
Views
3K
Undergrad Different norms and how L1 leads to the sparsest solution
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Is the Quotient of a Banach Space by a Closed Linear Subspace also Banach?
  • · Replies 1 ·
Replies
1
Views
3K
Insights Why Vector Spaces Explain The World: A Historical Perspective
  • · Replies 0 ·
Replies
0
Views
8K
High School Cartesian Space vs. Euclidean Space
  • · Replies 10 ·
Replies
10
Views
2K
Show that a space is a Banach space
  • · Replies 12 ·
Replies
12
Views
2K
High School What is a closed form solution?
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Open balls dense in closed balls in Euclidean space
  • · Replies 2 ·
Replies
2
Views
2K