Timeline for Why do infinite-dimensional vector spaces usually have additional structure?
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| when toggle format | what | by | license | comment | |
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| Aug 19, 2023 at 11:32 | vote | accept | Joe Lamond | ||
| Aug 17, 2023 at 15:55 | answer | added | wlad | timeline score: 3 | |
| Aug 17, 2023 at 15:46 | comment | added | Timothy Chow | @MikhailKatz Guilty as charged! :-) | |
| Aug 17, 2023 at 15:45 | comment | added | Mikhail Katz | @TimothyChow, you appear to be an avid reader of Penelope Maddy :-) | |
| Aug 17, 2023 at 14:53 | comment | added | Timothy Chow | (continued) When the question is phrased this way, we see that the answer is to be sought by looking at the examples and theorems we are most interested in. Often, in the examples of greatest interest, the questions we care about most can't even be phrased without extra structure. | |
| Aug 17, 2023 at 14:50 | comment | added | Timothy Chow | Here's another perspective. From textbooks, we can get the impression that mathematical objects begin life as structureless sets, and acquire more structure as we add it. But in "real life," we usually start with examples, which come with oodles of structure, and much of our task as mathematicians is to strip away much of that structure in order to clarify certain logical relationships. From this perspective, the question isn't why infinite-dimensional vector spaces have additional structure. The question is why we can't strip away all the structure and still deduce the theorems we want. | |
| Aug 17, 2023 at 7:53 | comment | added | Peter LeFanu Lumsdaine | @ Joe: following up @TomGoodwillie’s comment, don’t underestimate how much the same asymmetry you mention really does show up already in sets. Almost all uncountable sets arising in nature (the reals, sets of infinite sequences, sets of functions…) do naturally carry extra structure, most often some topology or similar, and the vector space examples you give seem very much linked to that. The idea that these sets “naturally come with” those topologies is well-studied in constructive mathematics (besides other areas) — e.g. work of Martín Escardó and collaborators. | |
| Aug 17, 2023 at 3:06 | comment | added | Kevin Casto | A refinement is to note that most infinite-dimensional vector spaces one encounters have uncountable dimension, e.g. all Banach spaces, and to reduce to the simpler question of "Why do uncountable sets in ordinary math (outside set theory) usually have extra structure?" | |
| Aug 17, 2023 at 2:00 | answer | added | A beginner mathmatician | timeline score: 4 | |
| Aug 17, 2023 at 0:23 | answer | added | Timothy Chow | timeline score: 15 | |
| Aug 16, 2023 at 22:32 | history | edited | Joe Lamond |
Added the general mathematics tag (feel free to remove if you don't think it is appropriate)
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| Aug 16, 2023 at 22:09 | history | became hot network question | |||
| Aug 16, 2023 at 21:59 | answer | added | user19232801 | timeline score: 18 | |
| Aug 16, 2023 at 21:19 | answer | added | Jochen Wengenroth | timeline score: 6 | |
| Aug 16, 2023 at 16:51 | comment | added | LSpice | Re, you are definitely using an inner product when you speak of rotations! Otherwise, where do the angles come from? | |
| Aug 16, 2023 at 15:49 | comment | added | Tom Goodwillie | You might just as well ask "Why do infinite sets usually have extra structure?" | |
| Aug 16, 2023 at 15:48 | answer | added | Mikhail Katz | timeline score: 10 | |
| Aug 16, 2023 at 15:45 | answer | added | ifatfirstyoudontsucceed | timeline score: 9 | |
| Aug 16, 2023 at 15:10 | comment | added | Joe Lamond | @user479223: This is a useful perspective, thank you. However, I still find the asymmetry a little puzzling, in light of the fact that some applications of finite-dimensional spaces seem to not use this topology. For instance, as far as I know, modelling linear transformations such as rotations in $\mathbb R^n$ does not require us to use its topology (or inner product, or norm). | |
| Aug 16, 2023 at 15:02 | comment | added | user479223 | I think one crucial point you're missing is that finite dimensional vector spaces implicitly come with extra structure. There is only one Hausdorff topology compatible with each finite dimensional vector space. All norms are equivalent etc. The topology is somehow baked in. The only difference is in infinite dimensions you must make a choice. | |
| Aug 16, 2023 at 14:16 | comment | added | Sam Hopkins♦ | What if "infinite-dimensional vector spaces with extra structure" don't occur more often per se, but there's so little general theory for infinite-dimensional vector spaces without extra structure that only the ones with structure are observed & studied. | |
| Aug 16, 2023 at 14:07 | history | asked | Joe Lamond | CC BY-SA 4.0 |