Form fields or vector fields over a manifold (as opposed to forms and vectors) do not form a vector space. They form a module.

The difference is that the scalars of a vector space form an algebraic field while the scalars of a module form a ring. For us humans (as opposed to "those higher beings that I do not understand (a.k.a. mathematicians)") this means that the scalars in the vector field can divide each other while the scalars in the spaces spanned by fields (i.e. a module) can not.

And just so we all can become even more confused: This has nothing to do with the fact that the "components" of each form field or vector field in certain basis are functions, i.e. themselves elements of a vector space with infinite number of dimensions.

The first way to see this module-not-a-vector-space characteristic is by
showing directly that the scalars that form the "coordinate components"
of a vector field can not always be divided, even if they are **not**
identically zero. Take, for instance the, manifold $
\mathbb{R}^2$ with the polar coordinate system and look at the vector
$ \begin{bmatrix} r \ r\cos(\theta) \end{bmatrix}$. The
"scalars" are $ r$ and $ r\cos(\theta)$. Obviously we
can not divide the former by the latter because it will be undefined at
$ \theta=\frac{\pi}{2}+n\pi$.

Another, more amusing way to show that the space spanned by these fields is not a vector space is to explicitly show that a property expected from vector spaces is not fulfilled. Namely, that in $ n$ dimensions an $ n$-uple of linearly independent elements forms a basis. However, in the case of fields over a manifold we can easily have a number of fields that are linearly independent over the manifold as a whole, and are at the same time linearly dependent (or simply equal to zero) on a subdomain. Hence, we have an $ n$-uple of linearly independent fields that can not be linearly combined to represent another arbitrary field.