I'm trying to understand one-forms in the context of general relativity. Lee (Introduction to Smooth Manifolds) says that at a point $p$ and with a vector field $X$ we define a covector field $df$, called the differential of $f$, by$$df_{p}\left(X_{p}\right)=X_{p}f.$$
My question is, is that definition true for any basis or only for a coordinate basis?
I understand that if $\omega^{a}$ is the one-form basis and $e_{b}$ is the vector basis then, again by definition, $$\omega^{a}e_{b}=\delta_{b}^{a}.$$ I also understand that using coordinate bases $\omega^{a}=\mathrm{d}x^{a}$ and $e_{b}=\partial_{b}$ then (from the first definition) $$\mathrm{d}x^{a}(\partial_{b})=\partial x^{a}/\partial x^{b}=\delta_{b}^{a}.$$ Thanks.