I am reading an introduction to differential geometry in the book "Classical Dynamics: a contemporary approach" by Eugene and Saletan (page 135).
Consider a manifold $Q$ of dimension $n$. They define a one-form $\alpha$ on $TQ$ as follows:
Let $X$ be a vector field on $TQ$ (I understand this as the vector calculus approach. for each point $q\in Q$ there exists a vector associated with it, and the collection of all the vectors form the vector field), then $\alpha(X)$ is a function on $TQ$. Also one-forms are linear: if $X,Y$ are two vector fields on $TQ$ and $f,g$ are two functions on the function space on $TQ$, then:
$$\alpha(fX+gY) = f\alpha(X)+g\alpha(Y).$$
I don't understand a couple of things: what does it mean to multiply a vector field and a function? And what does a function times a function mean? ($\alpha(X)$ is a function, and $f$ is a function, so what does $f\alpha(X)$ mean?).
I appreciate your help.
Given two real-valued functions $f, g : X \to \mathbb{R}$, their product is a real-valued function $fg : X \to \mathbb{R}$ given by $(fg)(x) = f(x)g(x)$; that is, $(fg)(x)$ is the product of the two real numbers $f(x)$ and $g(x)$.
Given a real-valued function $f : X \to \mathbb{R}$ and a function $g : X \to V$ taking values in a real vector space $V$, their product is a function $fg : X \to V$ taking values in $V$ which is given by $(fg)(x) = f(x)g(x)$; that is, $(fg)(x)$ is the product of the real number $f(x)$ and the vector $g(x)$.
Note, a vector field on a manifold is not simply a function taking values in a vector space, it is a function taking values in a family of vector spaces in a particular way (saying what this means precisely is equivalent to defining the notion of a section). Regardless, multiplying a vector field $\sigma$ on a manifold by a real-valued function $f$ has the same meaning: at each point $p$, you multiply the real number $f(p)$ and the vector $\sigma(p)$ to obtain a new vector $(f\sigma)(p)$.