In article <408EC9A3.DB73D12C@SPAMbtinternet.com.invalid>, George Cox <george_coxANTI@SPAMbtinternet.com.invalid> wrote:
>What is the precise >definition? If the precise definition has words like "abelian group" in >it, then what is the precise meaning of them? And so on.
Yeah, sure. A vector space is a module over a field. That's absolutely true and probably given that way in Bourbaki somewhere, but even in selective institutions I can't believe students learn about modules before vector spaces. Is there really a source anywhere that pretends to _introduce_ vector spaces in terms of abelian groups?
Hmm, I suppose some here would argue that it messes up students' understanding of modules if they are first tainted by facts learned for vector spaces, which then have to be unlearned in the general case.