Timeline for Verifying the Cauchy behavior of a sequence
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 18 at 18:13 | comment | added | PPB | Indeed, I am talking about the convergence of $\{x_n\}$ here. You are right, sir. It may need some other criteria to be fulfilled for the convergence of $\{x_n\}$. | |
| Mar 18 at 13:09 | comment | added | Willie Wong | Also, any particular reason you write "$\{x_n\}$ is Cauchy" and not that it converges? You are working in a Banach space after all. | |
| Mar 18 at 13:08 | comment | added | Willie Wong | I feel that this question is too broad. Certainly you have sufficient conditions like contraction mapping. But since your conclusion only concerns the values of $T$ on a countable subset, it certainly cannot be used to constrain $T$ in the general situation (so at the level of generalities I don't see any hope for a necessary condition). | |
| Mar 18 at 8:17 | history | edited | PPB | CC BY-SA 4.0 |
added 210 characters in body
|
| Mar 18 at 8:15 | comment | added | PPB | Also, I have made some changes to clarify my question. Thank you. | |
| Mar 18 at 8:12 | history | edited | PPB | CC BY-SA 4.0 |
added 210 characters in body
|
| Mar 18 at 8:09 | comment | added | PPB | @Willie Wong, my question is: Under what assumptions on T or any other hypothesis does the sequence $\{x_n \}$ become Cauchy? | |
| Mar 18 at 6:57 | comment | added | Pietro Majer | Maybe assumptions on K are in order | |
| Mar 18 at 6:12 | comment | added | Willie Wong | As stated, no. You can take $X = \ell_2(\mathbb{N})$. Then $W(x,y) = \|x - y\|^2$. Let $T$ be the shift map, and choose $x_1 = (1,0,0,\ldots)$. Then $x_n$ is not Cauchy. // But I wonder whether the question you typed above is actually the question you want to ask. Please double check and revise. | |
| Mar 18 at 3:32 | history | asked | PPB | CC BY-SA 4.0 |