Vector space of vector fields

Question

Is the set of all vector fields (on $\Bbb R^3$ for instance) a vector space? What about the set of all continuous vector fields on $\Bbb R^3$? It seems that this is just a generalization of the space of continuous functions from $\Bbb R$ to $\Bbb R$ which I know from linear algebra is a vector space. But I don't know if there is something that might go wrong (non-convergence or something) that would make the set of (continuous) vector fields not a vector space.

Answer 1

Yes, the set of vector fields on $\mathbb{R}^3$ forms a vector space with the obvious addition rule, ie. $$(A+B)(x,y,z)=A(x,y,z)+B(x,y,z)$$ and the obvious scalar multiplication. It is easy to check that this satisfies all the vector-space axioms. The set of continuous vector fields are closed under this form of addition and scalar multiplication (why?), and hence it forms a subspace.