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> for example, that the set of all vector spaces is itself a vector space Hmm... under what operations? It's a semigroup under direct sum, and probably somethi
by gunnihinn 10y ago
> for example, that the set of all vector spaces is itself a vector space
Hmm... under what operations? It's a semigroup under direct sum, and probably something under the tensor product, but I'm having trouble imagining what the scalar multiplication should be. (Or was that a "fictional" example? :)
- woopwoop 10y agoUnless I'm mistaken, The totality of vector spaces is not even a set.
- gunnihinn 10y agoThere's one for every cardinality (up to isomorphism). I forget my set theory, but if the cardinalities form a set, then the totality of vector spaces does too. If not, we can amuse ourselves by looking at the set of finite-dimensional vector spaces.
- wutbrodo 10y agoSigh, teach me to write a comment quickly without proofreading on HN, on a topic of any complexity. As written, my comment is wrong: I meant to write "vector space functions"[1] but left out a word. What I was getting at was the unintuitiveness that arises from thinking of vector space elements as number-like things (in the sense that we're familiar with them from basic math: numbers or coordinates) to being "anything that meets the definition", including functions. Seriously though, thanks for pointing that out, part of what I still love about the HN community is getting called out for even esoteric incorrectness. [1] which was my concise but imprecise way of expressing "the set of functions from a set over a field to a vector space over the same one", with + and * defined as you'd expect.