Timeline for What is the weak limit of $f_n \ \mathrm{sign}(f_n - 1)$ if $f_n \to f$ weakly in $L^p([0,1])$?
Current License: CC BY-SA 4.0
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| Aug 15, 2020 at 23:16 | comment | added | Lao | @ChristianRemling I see: for example, what assumptions do I need to add to obtain $f_{n_k}\mathrm{sign}(f_{n_k}-1) \rightharpoonup f \chi$, where $$\chi \in \begin{cases} 1 \text{ if } f > 1 \\ [-1,1] \text{ if } f = 1 \\ -1 \text{ if } f < 1 \end{cases},$$ weakly in $L^p$? | |
| Aug 15, 2020 at 18:09 | comment | added | Pietro Majer | The question may be reformulated e.g. as: what is the closure of the graph of the map $f\mapsto f\mathrm{sign} (f-1)$ from $L^\infty[0,1]$ to itself, wrto a given topology | |
| Aug 15, 2020 at 16:02 | comment | added | Christian Remling | It depends. For example, if $f_n\to 1$, then we can obviously have $g_n\to 1$ as well, but also $g_n\to 0$ if $f_n-1$ changes sign frequently. | |
| Aug 15, 2020 at 9:23 | history | edited | Lao |
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| Aug 15, 2020 at 8:49 | history | asked | Lao | CC BY-SA 4.0 |