Timeline for Why do infinite-dimensional vector spaces usually have additional structure?
Current License: CC BY-SA 4.0
6 events
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| Feb 16 at 14:13 | comment | added | Timothy Chow | @Joe I ran across a related MO question that you might find interesting: A vector space has the same dimension as its dual if and only if it is finite dimensional. | |
| Aug 17, 2023 at 11:38 | comment | added | Timothy Chow | @Joe It is certainly possible in principle for an "infiniteness hypothesis" to prevent pathological behavior. For example, a vector space cannot be the union of finitely many proper subspaces, if the field is infinite. (This fact comes up in the context of prime avoidance, which is a useful tool in commutative algebra.) But I think this sort of thing is rare, not just in linear algebra, but in mathematics more generally. I like your comparison with commutativity. | |
| Aug 17, 2023 at 10:14 | comment | added | Joe Lamond | I think this is similar to the point of view you are advocating in your answer, but I don't want to put words in your mouth, so I'd be interested to hear how you would answer the objection. | |
| Aug 17, 2023 at 10:14 | comment | added | Joe Lamond | Moreover, it is arguably quite unnatural to hypothesise that a vector space is infinite-dimensional; rather, we should hypothesise that it is not necessarily infinite-dimensional (which is equivalent to just studying general vector spaces, of course). I think this is similar to how many theorems about noncommutative rings are not really about rings which are noncommutative per se, but rather rings which are not necessarily commutative. | |
| Aug 17, 2023 at 10:13 | comment | added | Joe Lamond | Thanks for this answer. I think it is useful to think of finite-dimensionality as being the extra structure. One might object that infinite-dimensionality is also "extra structure" to the extent that it tells us something about the cardinality of a basis. One answer to this objection I came up with was the following: finiteness hypotheses are very useful in preventing pathological behaviour; however, the same cannot be said for infiniteness hypotheses. | |
| Aug 17, 2023 at 0:23 | history | answered | Timothy Chow | CC BY-SA 4.0 |