You must also add the condition that your scalars are real numbers. There are many finite dimensional vector spaces that are nothing like Euclidean space: any finite extension of a finite field, for example. This comes from Galois theory, but is not just abstract nonsense. One application to computing is if your scalars are only 0 or 1 and all operations are done mod 2, then you have a framework for doing error correcting codes, among other things. We call this set of scalars either the finite field of size 2 or the Galois field of size 2, aka GF(2).
Yes, what I should have said is: if k is an algebraically closed field then n-dimensional vector spaces over k are isomorphic to k^n. Unless I've had too much to drink this evening.
