Proof that closure of a space equals another space.

In summary, the homework statement is trying to prove that c0 is closed in ell0. The attempt at a solution shows that c0 is closed in ell0, but the problem is proving that elements in c0 have a vanishing limit.
  • #1
Paalfaal
13
0

Homework Statement



Define:

- c0 = {(xn)n [itex]\in[/itex] [itex]\ell[/itex][itex]\infty[/itex] : limn → [itex]\infty[/itex] xn = 0}

- l0 = {(xn)n [itex]\in[/itex] [itex]\ell[/itex][itex]\infty[/itex] : [itex]\exists[/itex] N [itex]\in[/itex] the natural numbers, (xn)n = 0, n [itex]\geq[/itex] N}Problem: Prove that [itex]\overline{\ell}[/itex]0= c0 in [itex]\ell[/itex][itex]\infty[/itex]

Homework Equations

The Attempt at a Solution



I want to find the solution using the limit-definition of closure.

Considering an element

x = (xn) [itex]\in[/itex] c0

and a sequence

yj = (xjn) [itex]\in[/itex] [itex]\ell[/itex]0,

such that xnj = xn for n < j, xnj = 0, otherwise.

Using the metric induced by the supremum norm; || xn - xnj || [itex]\rightarrow[/itex] 0 as j tends to infinity. We can du this for all elements in c0, and hence c0 [itex]\subseteq[/itex] [itex]\overline{\ell}[/itex]0.

My problem is to show the other direction, that is

[itex]\overline{\ell}[/itex]0 [itex]\subseteq[/itex] c0

I need to show that elements in [itex]\overline{\ell}[/itex]0 has a vanishing limit. I don't know how to do this using the supremum norm. In fact, it seems impossible to me..

Can I get any help?
 
Physics news on Phys.org
  • #2
Clearly, we know that [itex]\ell_0\subseteq c_0[/itex]. Try to prove now that [itex]c_0[/itex] is closed. This would imply what you try to prove.
 
  • #3
Hmmm.. When is it true that [itex]A \subseteq B \Rightarrow[/itex] [itex]\bar{A}[/itex] [itex]\subseteq[/itex] [itex]\bar{B}[/itex] ?


I ask because I've been working on a similar problem: Prove that [itex]\bar{\ell}[/itex]0 = [itex]\ell[/itex]2 in [itex]\ell[/itex]2.

With a similar approach of the attempt above I got [itex]\bar{\ell}[/itex]0 [itex]\supseteq[/itex] [itex]\ell[/itex]2.

Since [itex]\ell[/itex]0 [itex]\subseteq[/itex] [itex]\ell[/itex]2 (for obvious reasons),
[itex]\bar{\ell}[/itex]0 [itex]\subseteq[/itex][itex]\ell[/itex]2 iff [itex]\ell[/itex]2 = [itex]\bar{\ell}[/itex]2 (correct me if I'm wrong here).

In that case, [itex]\bar{\ell}[/itex]0 = [itex]\ell[/itex]2 in [itex]\ell[/itex]2.


(I guess it might is trivial that [itex]\ell[/itex]2 = [itex]\bar{\ell}[/itex]2 in [itex]\ell[/itex]2, but Is there an easy way to prove this?)

Any comments?

:smile:
 
  • #4
Paalfaal said:
Hmmm.. When is it true that [itex]A \subseteq B \Rightarrow[/itex] [itex]\bar{A}[/itex] [itex]\subseteq[/itex] [itex]\bar{B}[/itex] ?

This is always true. Try to prove it.

(I guess it might is trivial that [itex]\ell[/itex]2 = [itex]\bar{\ell}[/itex]2 in [itex]\ell[/itex]2, but Is there an easy way to prove this?)

If X is a metric space, then [itex]\overline{X}=X[/itex] (closure in X). Try to prove this.
 
  • #5
micromass said:
Try to prove it.

I certiantly will!


Thank you for your help! Much appreciated :smile:
 

1. What is meant by "closure of a space"?

The closure of a space, also known as the topological closure, is the set of all points that are either contained in a given space or are limit points of that space.

2. How is the closure of a space different from the space itself?

The closure of a space may contain additional points that are not included in the original space. These points are known as limit points and are necessary for defining continuous functions and other topological properties.

3. How can one prove that the closure of a space equals another space?

To prove that the closure of a space equals another space, one must show that every point in the closure is a limit point of the given space, and that every limit point of the space is contained in the closure. This can be done using various topological properties and definitions.

4. Why is the concept of closure important in topology?

The concept of closure is important in topology because it allows for the definition of continuous functions, compactness, connectedness, and other topological properties. It also helps to define the boundary and interior of a space, which are important concepts in many areas of mathematics and science.

5. Can the closure of a space be equal to the space itself?

Yes, the closure of a space can be equal to the space itself if the space is already closed. A space is considered closed if it contains all of its limit points. In this case, the closure of the space will simply be the space itself, as there are no additional limit points to include.

Similar threads

  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
15
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
28
Views
2K
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
2
Replies
37
Views
5K
Back
Top