Proving Open and Closed Sets for Sequence Spaces

In summary, we have discussed three questions concerning proving open and closed sets for specific sequence spaces. We have shown that the set A in question 1 is open by finding a radius δ that encloses a ball around each point in the set. We have also proven that c_0 is closed by showing that its complement is open in question 2. Finally, in question 3, we have shown that the set B is not open by finding a point in the set that does not have an open ball of some radius centered around it which is completely contained in the set.
  • #1
elias001
12
0
I have 3 questions concerning trying to prove open and closed sets for specific sequence spaces, they are all kind of similar and somewhat related. I thought i would put them all in one thread instead of having 3 threads.

1) Given y=(y[itex]_{n}[/itex]) [itex]\in[/itex] H[itex]^{∞}[/itex], N [itex]\in[/itex]N and ε>0, show that the set A={x=(x[itex]_{n}[/itex])[itex]\in[/itex] H[itex]^{∞}[/itex]:lx[itex]_{k}[/itex]- y[itex]_{k}[/itex]l<ε, for k=1,2,...N} is open in H[itex]^{∞}[/itex]

2) Show that c[itex]_{0}[/itex] is a closed subset of l[itex]_{∞}[/itex] [Hint: if (x[itex]^{(n)}[/itex]) is a sequence (of sequences!) in c[itex]_{0}[/itex] converging to x [itex]\in[/itex] l[itex]_{∞}[/itex], note that lx[itex]_{k}[/itex]l [itex]\leq[/itex] lx[itex]_{k}[/itex] - x[itex]^{n}_{k}[/itex]l + lx[itex]^{n}_{k}[/itex]l and now choose n so that lx[itex]_{k}[/itex] - x[itex]^{n}_{k}[/itex]l is small independent of k.]

3) show that the set B={x [itex]\in[/itex] l[itex]_{2}[/itex]: lx[itex]_{n}[/itex]l[itex]\leq[/itex]1/n, n=1,2,..} is not an open set. [hint: is the ball B(0,r) a subset of B.]

for question 1
The metric for H^infinity is A,

d(x,y)=Ʃ[itex]^{∞}_{i=1}[/itex]2[itex]^{-n}[/itex]lx[itex]_{n}[/itex]-y[itex]_{n}[/itex]l

so if m is in A, then I can enclosed a ball B(m,[itex]\delta[/itex]) of radius delta around m where [itex]\delta[/itex]=M[itex]_{k}[/itex]+1/2^k where M[itex]_{k}[/itex]=max{lx[itex]_{k}[/itex]-y[itex]_{k}[/itex]l, k=1,..,N} would that work?

For question 2,

in the hint, where it says to choose n, so that... how do i pick n so that lx[itex]_{k}[/itex] - x[itex]^{n}_{k}[/itex]l is small.

I know that lx[itex]_{k}[/itex]l≤ lxl, but you have a sequence x[itex]^{n}_{k}[/itex] converging to x, is d(x[itex]^{n}_{k}[/itex], x) ≥ d(x[itex]^{n}_{k}[/itex], x[itex]_{k}[/itex])?? If so, how can i use this fact to pick n?

for question 3.

From the hint where it asks whether the open ball of radius r around 0 is a subset of B. In this case, would i have to specific how large r would be so that the radius of B(0,r) would contains elements that are not in B. If so, how do i specific r. The minkowski inequality is not of any help.
 
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  • #2
Can someone please help me out. 1) For the first question, the idea is to show that the set A is open by showing that for each point in the set there is an open ball of some radius centered around the point which is contained entirely in the set. In this case, since d(x,y) is defined as the sum of the absolute values of the differences between the two sequences, you can find a radius δ such that |x_k-y_k|<ε for k=1,...,N and 0 for k>N. This means that if we choose δ = max{|x_k−y_k|:k=1,...,N} + 1/2^k, then any point within the ball B(m,δ) will have |x_k−y_k|<ε for all k∈N, and thus be in A. This shows that A is open.2) For the second question, the idea is to show that c_0 is closed by proving that its complement is open (i.e., its complement is the union of open sets). In this case, the complement is {x∈l_∞: lim_{n→∞} x_n ≠ 0}. So for each x∈l_∞, we can find an open ball B(x,r) such that any point in the ball has a limit that differs from x. Thus, the complement of c_0 is the union of these open balls, and thus is open. 3) For the third question, the idea is to show that B is not open by finding a point in B that does not have an open ball of some radius centered around it which is completely contained in the set. To do this, consider the point x=(1/n ) where n∈N. This point is clearly in B. Now, consider the open ball B(x,r) around x. Any point in the ball will have lx_k l ≤ r + 1/n, so it is not in B unless r ≥ 1/n. Thus, there is no open ball centered around x which is completely contained in B, so B is not open.
 

1. What is the definition of an open set?

An open set is a set that contains all of its limit points. In other words, for every point in an open set, there exists a small enough neighborhood around that point that is also contained within the set.

2. How do you prove that a set is open?

To prove that a set is open, you must show that for every point in the set, there exists a neighborhood around that point that is also contained within the set. This can be done by choosing a specific radius for the neighborhood and showing that all points within that radius are also in the set.

3. What is a closed set?

A closed set is a set that contains all of its limit points as well as its boundary points. In other words, the set is closed off and does not have any gaps or holes.

4. How do you prove that a set is closed?

To prove that a set is closed, you can use the sequential characterization of closed sets. This means that for every convergent sequence within the set, the limit of the sequence must also be within the set. If this condition is met, then the set is considered closed.

5. What are some common examples of open and closed sets in sequence spaces?

In sequence spaces, common examples of open sets include sets of sequences that converge to a specific limit, such as all sequences that converge to 0. Examples of closed sets in sequence spaces include sets of sequences that do not converge, such as all sequences that do not have a limit or diverge to infinity.

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