Timeline for Pontryagin-reflexivity of spaces of continuous functions
Current License: CC BY-SA 4.0
7 events
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| Mar 26, 2022 at 10:53 | comment | added | user103549 | I didn't object to the use of contractions--why would I? I just had the impression that the duality you describe is different from the one mentioned in the question, but as you didn't go into detail, I can't really tell. | |
| Mar 26, 2022 at 8:10 | comment | added | memorial | By the way, I think that your objection to the use of contractions is a bit of a red herring. They were only used as a a framework to place the extensions of duality from the special cases (finite dimensional, compact, etc.) in a unified setting. This is an a posteriori approach--all of these extensions were made previously without category theory and there is nary a word of contractions in the final version. | |
| Mar 26, 2022 at 8:09 | comment | added | memorial | I would disagree with the comment that the Saks space structure doesn´t come from a natural mapping space topology (but that´s just business, not personal). If you want to work with compact convergence, the way to go would be to extend the symmetric duality between Banach spaces and Waelbroeck spaces to the corresponding pro and ind completions. The former is the category of (complete) locally convex spaces, the latter a special class of convex bornological spaces, where the balls are provided with a suitable compact topology. The latter has never been studied, to my knowledge. | |
| Mar 26, 2022 at 7:23 | comment | added | user103549 | Concerning the last paragraph, if you're sure I can't just read it up somewhere, I would probably bother to "sit down and do this". ;) (One last note: I'm explicitly not considering just bounded functions, but the exponential $\mathbb{R}^X$.) And thanks again for taking your time to answer. | |
| Mar 26, 2022 at 7:18 | comment | added | user103549 | Thank you for your answer! The first paragraph does not seem to be about what I am asking, though, since I am considering Banach spaces as topological vector spaces here, and not with short linear maps. As for the second paragraph, here is a list of things that I am not aiming for: Saks spaces; convergence spaces and c-reflexivity (Butzmann et al.); stereotype spaces (Akbarov). The all have nice duality theories for spaces of continuous functions, but they don't come from natural mapping space topologies, as described in the question. | |
| Mar 25, 2022 at 12:25 | history | edited | memorial | CC BY-SA 4.0 |
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| Mar 25, 2022 at 12:15 | history | answered | memorial | CC BY-SA 4.0 |