Talk:Seminorm: Difference between revisions
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== Field of Scalars == |
== Field of Scalars == |
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Is there any reason we restrict to vector spaces over '''R''' or '''C'''? The definition makes sense for vectors spaces over any valued field, right? [[User:ATC2|ATC2]] ([[User talk:ATC2|talk]]) 10:59, 22 April 2023 (UTC) |
Is there any reason we restrict to vector spaces over '''R''' or '''C'''? The definition makes sense for vectors spaces over any [[valued field]], right? [[User:ATC2|ATC2]] ([[User talk:ATC2|talk]]) 10:59, 22 April 2023 (UTC) |
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:Interesting question. Yes, feels like that should be true. Almost surely almost everywhere this is true. However, this article is just large enough that one would have to comb over it, and ask, for each statement, "is this still true for a valuation"? Lets suppose it is. And you might know that it is, but then some younger student clicks through to the cited book/chapter, and will be surprised to find that it talks about reals/complex and not general valued fields. And so how can one know for sure? Well, there are several projects that attempt to codify all math theorems into machine-provable form, and you could use them as an oracle: "is it true that X?". Suppose that this oracle delivers "yes" on every statement in this article. Now what? Should we cite that oracle? Someone will claim that this is [[WP:OR]]! I don't think it is, its "too obvious" to be OR. So, yes, I think its true, and its an interesting question. [[Special:Contributions/67.198.37.16|67.198.37.16]] ([[User talk:67.198.37.16|talk]]) 19:46, 27 May 2024 (UTC) |
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Latest revision as of 19:46, 27 May 2024
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Field of Scalars
[edit]Is there any reason we restrict to vector spaces over R or C? The definition makes sense for vectors spaces over any valued field, right? ATC2 (talk) 10:59, 22 April 2023 (UTC)
- Interesting question. Yes, feels like that should be true. Almost surely almost everywhere this is true. However, this article is just large enough that one would have to comb over it, and ask, for each statement, "is this still true for a valuation"? Lets suppose it is. And you might know that it is, but then some younger student clicks through to the cited book/chapter, and will be surprised to find that it talks about reals/complex and not general valued fields. And so how can one know for sure? Well, there are several projects that attempt to codify all math theorems into machine-provable form, and you could use them as an oracle: "is it true that X?". Suppose that this oracle delivers "yes" on every statement in this article. Now what? Should we cite that oracle? Someone will claim that this is WP:OR! I don't think it is, its "too obvious" to be OR. So, yes, I think its true, and its an interesting question. 67.198.37.16 (talk) 19:46, 27 May 2024 (UTC)