In page 138 in John Lee's Introduction to Smooth Manifolds, he stated the following:
"We begin with the simplest case: an interval in the real line. Suppose $[a,b] \subset \mathbb{R}$ is a compact interval, and $\omega$ is a smooth covector field on $[a,b]$. If we let $t$ denote the standard coordinate on $\mathbb{R}$, $\omega$ can be written $\omega_t = f(t)dt$ for some smooth function $f: [a,b] \rightarrow \mathbb{R}$."
I don't get why $\omega$ can be written $\omega_t = f(t)dt$. From what I understand, I have $$ \omega: [a,b] \rightarrow T^{\ast}([a,b])$$ $$t \mapsto \omega_t: T_{t}([a,b]) \rightarrow \mathbb{R}$$ As $T^{\ast}([a,b]) \cong [a,b]$, we have $\omega_t:[a,b] \rightarrow \mathbb{R}$. I am not sure how to carry on from here.