is the vector space of n- forms of an n-manifold equal to the vector space of compactly supported n-forms?

Question

Let $\Omega^{n}(M)$ be the real vector space of smooth n-forms of an n-manifold $M$. It is a real vector space of dimension 1. $\Omega^{n}_c(M)$ is the real vector space of compactly supported smooth n-forms on $M$. Since it is a subspace of $\Omega^{n}(M)$, which has dimension 1, shouldn't be equal to the whole $\Omega^{n}(M)$ or to $\{0\}$? But there are n-forms that are not compactly supported.

Answer 1

I think that, for the purposes of your confusion, we can first consider changing "n-form" to "vector field" in order to gain some more concreteness first. The confusion is the same.

Consider the set of vector fields on a manifold $M$, let's call it $V(M)$. It is the set of smooth sections of the tangent bundle $TM$. What we are doing is assigning to each point $x$ of $M$ a point on $T_xM$ in a "smooth" way. This is a real vector space, since we can add two vector fields: we do it by adding in each $T_xM$. The dimension of $T_xM$ is $n$, but not of $V(M)$. It is a different object.

Analogously, in your case, a smooth n-form is a section on the n-th exterior power of the cotangent bundle. In each point, you have a vector space of dimension $1$. But $\Omega(M)$ is not that vector space, it is a different object.