Timeline for Why is $\Xi \equiv 0$ if $E=C_{0}(X \times Y)$?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 14, 2022 at 10:51 | comment | added | Akira | Thank you so much for your help :v | |
| May 14, 2022 at 10:40 | comment | added | Dirk | Phew, I am not sure if I find the time and energy to chase down that detail… | |
| May 14, 2022 at 8:09 | comment | added | Akira | If you don't mind, please have a look at another closely related question here. | |
| May 13, 2022 at 23:22 | vote | accept | Akira | ||
| May 13, 2022 at 22:22 | comment | added | Akira | [...] Assume $X$ is not compact and $Y$ is compact. Let $(\varphi, \psi) \in C_0(X) \times C_0(Y)$ such that $\psi \equiv 0$. Define $u (x,y) := \varphi(x)+ \psi(y)$. Indeed, $u \in C_0(X \times Y)$. Fix $\varepsilon>0$. There is a compact set $K \subset X$ such that $\sup_{x \notin K} |\varphi(x)| \le \varepsilon$. Clearly, $K \times Y$ is compact. Also, $$ \sup \{ |u(x,y)| \mid (x,y) \notin K \times Y\} = \sup \{ |\varphi(x)| \mid x \notin K \} \le \varepsilon. $$ | |
| May 13, 2022 at 22:22 | comment | added | Akira | It seems the statement in the exercise holds only when $X,Y$ are both non-compact. Could you confirm if my observation is correct? [...] | |
| May 13, 2022 at 11:42 | history | answered | Dirk | CC BY-SA 4.0 |