Why Is This Set Closed in l2 Yet Lacks a Minimum Norm?

[smiley2]

Hellou!

I have a question regarding the square summable sequences:
I should find an example of a closed set from the square summable sequences and show that the closed set does not have an element with a min norm!
The professor mentioned an example:
(1+1/1 0 0 0...)
(0 1+1/2 0 0...)
...
(0...0 1+1/n 0...)

this set does not have an element with min norm, but why is it a closed set in l2 ?!

 

[Xevarion]

they are all isolated points.

 

[tacman]

Hello! In order to understand why this set is closed in l2, we need to first define what a closed set is in this context. A closed set in l2 is a set of square summable sequences where every limit point of the set is also a member of the set. In simpler terms, this means that if we take any sequence that is a limit of the set, it must also be a part of the set.

In the example given by your professor, the set consists of sequences with elements that are either 1 or 1+1/n. This set is closed because every limit point of this set, which is a sequence with elements that are either 1 or 1+1/n, is also a member of the set. For example, if we take the limit of the sequence (1, 1+1/2, 1+1/3, ...), it will converge to the sequence (1, 1, 1, ...), which is also a member of the set.

Now, why does this set not have an element with minimum norm? This is because the minimum norm of any element in this set is 1, but there is no element in the set that has all its elements equal to 1. Therefore, there is no element with minimum norm in this set.

I hope this helps clarify the concept of closed sets and why this particular set does not have an element with minimum norm. Let me know if you have any further questions.