Timeline for Takahashi minimization theorem for lower pseudo-continuous functions on complete metric spaces
Current License: CC BY-SA 4.0
5 events
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| Sep 16, 2022 at 13:18 | comment | added | Jochen Wengenroth | That's awesome. Nevertheless, I think that the standard proof (without transfinite induction goes through): One assumes that there is no minimizer and constructs recursively a sequence $x_n$ with $f(x_{n+1})\le (f(x_n)+c_n)/2$ where $c_n=\inf f(B_n)$ and $B_n=\{y\in X: f(y)\le f(x_n)-d(x_n,y)\}$. This sequence is Cauchy and lower semicontinuity is only used to show that the limit $x_*$ satisfies $f(x_*)\le \liminf\limits_{n\to\infty} f(x_n)$. But the sequence $f(x_n)$ is strictly decreasing so that the limit exists and the inequality follows from lower pseudocontinuity. | |
| Nov 1, 2018 at 12:43 | vote | accept | M. Reza. K | ||
| Oct 31, 2018 at 6:24 | history | edited | Taras Banakh | CC BY-SA 4.0 |
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| Oct 30, 2018 at 16:54 | history | edited | Taras Banakh | CC BY-SA 4.0 |
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| Oct 30, 2018 at 16:31 | history | answered | Taras Banakh | CC BY-SA 4.0 |