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Thanks Martin The lemma of Extension to the dimension one higher Is the result \hat{f}(x)=θ and \hat{f}(x)\leq p(x)? In your conclusion, why "Now if we apply Hahn-Banach theorem once again to fˆ, we get a functional LIM:ℓ∞→R which is a generalized limit and we also have" Why we have LIM(x_0)=\hat{f}(x_0)=\theta? Do you mean for each extension of \hat{f}, \hat{f}(x_0) always take same value \theta?
@MartinSleziak
Well, I am a little confused about your proof of question 1. I know your lemma shows that for the first extension(not the whole space) we still have the $\hat{f}$ is dominated and takes value \theta. But why we can apply H-B theorem again, state that $LIM$ still takes the same value $\theta$? Use the fact $\liminf \leq LIM\leq \limsup$ then we need to let $\theta \in [\liminf, \limsup]$?
Anther question:
For "If x is a sequence such that for any generalized limit we get the same value L". Do we need to use the result in part (1)? Another question, if I consider the Banach sequence which is defined by a generalized limit with shift-invariant. Is B(\{a_n\})=LIM(1/n \sum_{j=1}^n x_{j}) a Banach limit? Here is my proof:
Do you think if it is right? Here is the key point: LIM((\frac{x_1-x_{n+1}}{n})+B(x_2,x_3,x_4,\dots)=B(x_2,x_3,x_4,\dots) since $\Vert x_1-x_{n+1}\Vert \leq 2\Vert x\Vert_{\infty}$ which is bounded. Then LIM (\frac{x_1-x_{n+1}}{n})=0 Then I can show that $B(\{x_{n+1}\})=B(\{x_{n}\})$. Remark: I use B(\{x_{n}\})=B(x_1,x_2,x_3,\dots)=B(x_1-x_2, x_2-x_3,x_3-x_4,\dots)+B(x_2,x_3,x_4,\dots)=LIM(\frac{1}{n}\sum_{i=1}^n(x_i-x_{i+1}))+B(x_2,x_3,x_4,\dots) Denote the shift operator $T: l_{\infty}\to l_{\infty}$ by \[T: (x_1, x_2, x_3, \dots)\to (x_2, x_3, x_4, \dots).\] So we have $B(T\{x_{n}\})=B(\…
In case you have some other question from functional analysis. (Of course, we could discuss the stuff related to my answer here. But the question about Banach limits seems to be a new question.)
@TYG Yes. Because $\operatorname{LIM}|_{\widehat M}=\widehat f$ and $x\in\widehat M$.
BTW you're probably not using bookmarklet for MathJax in chat, I guess. (Judging by the fact that you omit dollars in many places - which makes your messages a bit harder to read.)
> It only remains to prove the shift-invariance. To show this, simply observe that $\frac{x_1+\ldots+x_n}n - \frac{(Sx)_1+\ldots+(Sx)_n}n = \frac{x_1+\ldots+x_n}n - \frac{x_2+\ldots+x_{n+1}}n= \frac{x_1-x_{n+1}}n$.
The body of the answer you linked says (emphasis mine):
It depends; Look at the context of how they are asked.
You'll want to keep the question **if the wording provides another way for a search query to find the content.** It's a bit like a "see also …" entry in an index.
Delete it if the duplicate does not add terminology or alternate phrasing to find the question. It may not be worth cluttering up the system with this exact duplicate. There are several poorly phrased questions that will not direct traffic to us.
We've come to rename Markdown to Letdown in the C++ chat room because it lets you down so often. I've now just found a pattern.
It seems markdown fails for multi-line messages. That is, this
Letdown can't cope with multi-line comments.
Let's see code?
fails to display code marked as co...
Is there a list of privileges that users have on any Stack Exchange chat rooms, and which lists the minimum reputation required for each privilege?
If such list doesn't exist, what is the reputation required for the chat privileges?
We've come to rename Markdown to Letdown in the C++ chat room because it lets you down so often. I've now just found a pattern.
It seems markdown fails for multi-line messages. That is, this
Letdown can't cope with multi-line comments.
Let's see code?
fails to display code marked as co...
