Show that $\omega$ is a $C^{\infty}$ - linear map

Question

Let $\omega \in \Omega^1(M)$ be a $1-$ form on a smooth manifold $M$. Let $\chi(M) $ be the vector space of $C^{\infty}$ vector fields on $M$. Then I need to show that $\omega : \chi(M) \rightarrow C^{\infty}(M) $ given as $\omega(X)(p) = \omega_p(X_p)$ ; $p \in M , X \in \chi(M)$ defines a $C^{\infty}$ - linear map. ($\omega_p$ is the co-vector at $p$).

Now I know that I need to show the following - $$\omega(fX + gY) = f \omega(X) + g \omega(Y)$$ for $X , Y \in \chi(M) $ and $f,g \in C^{\infty}(M)$.

But I am not getting any idea on how to prove the last expression.

Answer 1

You can calculate that expression to the point $p\in M$, so we have:

\begin{equation} \begin{split} \omega_p(fX + gY)_p =&\omega_p(f(p)X_p + g(p)Y_p)\\=&f(p)\omega_p(X_p) + g(p)\omega_p(Y_p)\\=&f(p)\omega(X)(p)+g(p)\omega(Y)(p) \end{split}\end{equation}

(I used the linearity of the functional $\omega_p$ in the second line), but this is true for any $p\in M$, so we have: \begin{equation} \omega(fX + gY) =f\omega(X)+g\omega(Y) \end{equation}