Show Co is Closed in L∞: Finding Ko

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In summary, we want to show that the Banach space co is closed in l∞ by showing that x is in co, and to do this, we need to find a Ko such that for all k>Ko, |x(k)|<e. Using the fact that x_n converges to x, we can find this Ko by choosing the largest value of k from the inequality |x_n(k) - x(k)| < e for all n>N_e.
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I want to show the Banach space co is closed in l∞ .

So, I pick a convergent sequence x_n in co that converges to x in l∞
Now, x_n --> x: given e>0, there is an N_e s.t. for all n>N_e,
||x_n -x ||= Sup |x_n(k)-x(k)|<e (we're supping over k).

Since x_n is a sequence in co , for each fixed n, x_n(k)-->0 as k--> infinity.
So, given e>0, there is a K depending on n and e, such that for all k> K, we have |x_n(k)|<e.

We want to show x is in co
so we show there is a Ko such that for all k>Ko, |x(k)|<e

I am having trouble getting this Ko.

I know |x(k)|≤ |x_n(k)| + |x_n(k)-x(k)|≤ |x_n(k)| + Sup (over k) |x_n(k)-x(k)|

we have |x_n(k)|<e as k>K, but Sup (over k) |x_n(k)-x(k)|<e for n>N_e.


So I am not so sure how to get this Ko.
 
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Hi there,

First of all, great job on starting the proof! To find the Ko that you need, let's think about what we know about the sequence x_n that converges to x.

We know that for any e>0, there exists an N_e such that for all n>N_e, ||x_n - x|| < e. In other words, as n gets larger and larger, the terms in the sequence x_n get closer and closer to the corresponding terms in x.

Now, let's focus on a specific term in x, say x(k). We want to show that there exists a Ko such that for all k>Ko, |x(k)| < e. In other words, we want to show that as k gets larger and larger, x(k) gets closer and closer to 0.

Since x_n converges to x, we know that for any e>0, there exists an N_e such that for all n>N_e, |x_n(k) - x(k)| < e. This means that as n gets larger and larger, the difference between x_n(k) and x(k) gets smaller and smaller.

So, to find our Ko, we want to find the largest value of k that we need to guarantee |x(k)| < e. This will be our Ko.

Since we know that |x_n(k) - x(k)| < e for all n>N_e, we can choose the largest value of k from this inequality. In other words, we can choose Ko = max{k | |x_n(k) - x(k)| < e for all n>N_e}.

I hope this helps and clarifies how to find the Ko that you need in your proof. Keep up the good work!
 

1. What is "Show Co is Closed in L∞: Finding Ko" about?

"Show Co is Closed in L∞: Finding Ko" is a scientific study that investigates the concept of closed sets in mathematical analysis. It specifically delves into the concept of closed sets in the topological space of L∞, which is the space of all bounded and measurable functions.

2. Why is the concept of closed sets important in mathematical analysis?

The concept of closed sets is crucial in mathematical analysis because it allows for the definition of important concepts such as continuity, compactness, and convergence. It also helps in the formulation of theorems and proofs in various branches of mathematics such as calculus, topology, and functional analysis.

3. What is L∞ and why is it used in this study?

L∞ is the space of all bounded and measurable functions, also known as the space of essentially bounded functions. It is used in this study because it is a complete metric space, which means that all Cauchy sequences in this space converge to a limit. Additionally, the space of L∞ is particularly useful in functional analysis and measure theory.

4. What is the significance of finding Ko in this study?

Finding Ko, or the Ko boundary, is important in this study because it allows for the identification of points in L∞ that are close to the boundary of closed sets. This can help in understanding the behavior of functions near these boundaries and can aid in proving theorems related to closed sets and their properties.

5. How can the findings from "Show Co is Closed in L∞: Finding Ko" be applied in other areas of science?

The results of this study can be applied in various areas of science, particularly in fields that utilize mathematical analysis and topological spaces. This includes physics, engineering, economics, and computer science, among others. The concept of closed sets and the techniques used in this study can provide valuable insights and aid in problem-solving in these fields.

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