Timeline for Pontryagin-reflexivity of spaces of continuous functions
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 26, 2022 at 7:56 | answer | added | Sergei Akbarov | timeline score: 2 | |
| Mar 26, 2022 at 7:00 | history | edited | user103549 | CC BY-SA 4.0 |
added 81 characters in body
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| Mar 26, 2022 at 6:56 | comment | added | user103549 | @Gro-Tsen If $X$ is a discrete space or more generally a disjoint union of compacta, $\mathbb{R}^X$ is indeed Pontryagin-reflexive, since (if you trust Wikipedia) Kaplan showed that arbitrary products of Pontryagin-reflexive groups are Pontryagin-reflexive. Although there's the caveat that he probably didn't use the product in $k$-spaces, but it probably works anyways, in this case. | |
| Mar 25, 2022 at 12:15 | answer | added | memorial | timeline score: 2 | |
| Mar 25, 2022 at 12:11 | comment | added | Gro-Tsen | At any rate, I would expect the answer to have much to do with realcompactness of $X$. Did you try to work out what happens if $X$ is a discrete set whose cardinality is a measurable cardinal? (See Gillman & Jerison, Rings of Continuous Functions for the general background on realcompact spaces.) | |
| Mar 25, 2022 at 12:09 | comment | added | Gro-Tsen | Maybe you should clarify what you mean by $\operatorname{Hom}_{\mathbb{R}}(—,\mathbb{R})$: continuous $\mathbb{R}$-linear maps? (Because the set of ring homomorphisms $\mathbb{R}^X \to \mathbb{R}$ is also important and much studied.) | |
| S Mar 25, 2022 at 11:39 | review | First questions | |||
| Mar 25, 2022 at 12:10 | |||||
| S Mar 25, 2022 at 11:39 | history | asked | user103549 | CC BY-SA 4.0 |