I'm reading Spivak's Calculus on Manifolds (I'm on page 89). For a differentiable function $f : \mathbb{R}^n \to \mathbb{R}^m$, $Df(p): \mathbb{R}^n \to \mathbb{R}^m$ is a linear transformation (on the tangent space at $p \in \mathbb{R}^n$, I believe); it is just the Jacobian matrix. Spivak then defines the following linear transformation between tangent space: $f_*: \mathbb{R}^n_p \to \mathbb{R}^m_{f(p)},$ as $$f_*(v_p) = (Df(p)(v))_{f(p)}.$$ Then, the linear transformation $f^*:\Lambda^k(\mathbb{R}^m_{f(p)}) \to \Lambda^k(\mathbb{R}^n_p)$ (Spivak doesn't name it, but I believe it is the pullback) is defined, where $\omega$ is a $k$-form on $\mathbb{R}^m$, by $$(f^* \omega)(p)(v_1, v_2, \dots, v_k) = \omega(f(p))(f_*(v_1), f_*(v_2), \dots, f_*(v_k)).$$
Then, Spivak presents a theorem which states that if $f:\mathbb{R}^n \to \mathbb{R}^m$ is differentiable, then $$f^*(g \cdot \omega) = (g \circ f) \cdot f^* \omega.$$
I don't see why this theorem is true. I think part of the reason I am confused is that I don't quite understand what $g \cdot \omega$ means. So in general, you can write $$\omega(p) = \sum_{i_1 < \dots < i_k} \omega_{i_1 \dots i_k}(p) \cdot \varphi_{i_1}(p) \wedge \dots \wedge \varphi_{i_k}(p), $$ where these $\varphi_{i_j}$ are the dual vectors for some basis vectors $v_{i_j}$ of $\mathbb{R}^n.$ I don't really understand what $g \cdot \omega$ means. Does it refer to the $k$ form $$\sum_{i_1 < \dots < i_k} g(p)\omega_{i_1 \dots i_k}(p) \cdot \varphi_{i_1}(p) \wedge \dots \wedge \varphi_{i_k}(p), $$ so you interpret it as the product of functions? Even if it is this, I still don't quite see why we should get a $g \circ f$ in the above theorem. Any advice would be great!