In proving $\mathfrak X(U)$ is a $C^{\infty}U$-module, for an open subset $U$ of $\mathbb R^n$ my book defines scalar multiplication of smooth vector fields $U$ by smooth functions on $U$ as
$$[fX]_p := f(p)X_p$$
I could not find a definition for addition. Is it $[X+Y]_p := X_p + Y_p$ ? Otherwise, with what definition do we prove the "Abelian group" part of showing $\mathfrak X(U)$ is a $C^{\infty}U$-module?
My book is An Introduction to Manifolds by Loring W. Tu.