Timeline for Quantifier swap in Banach space theory
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| May 28, 2020 at 18:20 | history | edited | Jason Zhao | CC BY-SA 4.0 |
typo, $f_i$ should be $f$
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| May 28, 2020 at 18:20 | comment | added | Jason Zhao | @AndrejBauer yes, thanks for catching the typo | |
| May 28, 2020 at 6:44 | comment | added | Jochen Glueck | It might be worthwhile to note that the very definition of a Banach space - i.e., completeness - can be phrased in terms of a quantifier swap: A metric space $(M,d)$ is complete if and only if, for every sequence $(x_n)$ in $M$, the following two assertions are equivalent: $\exists x \in M \; \forall \varepsilon > 0 \; \exists n_0 \; \forall n \ge n_0: \; d(x_n,x) < \varepsilon$, and $\forall \varepsilon > 0 \; \exists x \in M \; \exists n_0 \; \forall n \ge n_0: \; d(x_n,x) < \varepsilon$. (Since the second assertion is equivalent to $(x_n)$ being a Cauchy sequence.) | |
| May 28, 2020 at 6:17 | comment | added | Andrej Bauer | Should the last $f_i$ in the displayed formula be $f$? | |
| May 28, 2020 at 6:15 | comment | added | Andrej Bauer | See my answer to "Techniques for reversing the order of quantifiers". While you're reading it I'll try to see if your example is also about compactness. | |
| May 27, 2020 at 21:07 | comment | added | Cameron Zwarich | Is there a logical interpretation of other quantifier swaps in analysis, e.g. any of the more basic compactness arguments? | |
| May 27, 2020 at 20:14 | comment | added | Robert Israel | Swapping quantifiers depends a lot on the statement that comes after the quantifiers, and not just on the Banach space. Even in a finite set, $\forall x \exists y \; A(x,y)$ may or may not be equivalent to $\exists y \forall x \; A(x,y)$, depending on $A$. | |
| May 27, 2020 at 17:50 | review | First posts | |||
| May 27, 2020 at 18:20 | |||||
| May 27, 2020 at 17:45 | history | asked | Jason Zhao | CC BY-SA 4.0 |