>I was taught (33 years ago in the University of Birmingham, UK) that a >vector space is >(1) a set of elements (the vectors) which form an abelian group under >addition; >(2) etc.... >The phrase "commutative group" or "additive group" might have been used >instead of "abelian group"--you'll forgive me if I can't quite remember! >
I think you just made me remember what an abelian group is. It's closed under its operation, it's associative, commutative, it's got an inverse, and it's got the identity element in it. There may be some other condition, since aren't all the above needed for any group? I remember something about an abelian group forming a table with a diagonal, or something....
So, basically, "abelian group" is just an easier way of saying "vector space". Rather than listing all 8 conditions. But, that assumes the student *knows* what an abelian group is.
>How do you/US universities/today's commonly used texts define vector >space?