Product over a vector space

Question

When looking at the definition of a vector space, I see that it's basically a set with two operations and a set of 8 axioms.

However, none of those axioms talk about the product of two vectors.

Is that always defined in a vector space ? If so, how ?

Answer 1

Since all finite-dimensional vector spaces are isomorphic to $F^n$ for $F$ a field and $n \in \mathbb N$, it is always possible to define a multiplication operation pointwise, and get a ring.

However, just because it can be done, doesn't mean it should. This pointwise multiplication is not suitable for geometry, since it is not invariant under change of basis. It is not the "right" definition of multiplication of polynomials, in that it does not interact nicely with evaluation at a point.

Moreover, when defining maps between vector spaces, linear maps preserve addition and scaling but need not preserve either this multiplication or any other. For example, differentiation of polynomials (or indeed functions in general) preserves the vector space operations but is not a homomorphism for multiplication. So it's important to set aside the multiplication in certain contexts, even when you know it's there.

Post-script: what about infinite-dimensional vector spaces? Well, they are all isomorphic to the space of elements of finite support in $F^{|S|}$, where $S$ is some basis. Then you can well-order the basis and define a generalised polynomial multiplication on the coefficients. I'm pretty sure that puts a ring structure on any vector space, although it's usually an intractable one.

Answer 2

No, in general this is not defined.

You need something more to make this work.

As it stands vector spaces are Abelian groups (over addition). To add multiplication you need a ring structure on the set of vectors. Some vector spaces have this more or less naturally but some do not.