On page 90 of Spivak's Calculus on manifolds
4-8 Theorem (3) $f^*(g.w)=(g\circ f)\cdot f^*w $.
Here I assume $g $ is a function $$f^*(g.w)(p)(v_p)=(g.w)(f(p))(f_*v)=g(f(p))(f_*v)\cdot w(f(p))(f_*v)=(g \circ f_*)(f^*w) $$
What is wrong with my calculation?
Also, I want clarity on following questions
1) Is $w$ which is a $k-form$ a vector field ?
2)What sort of function is $g$ i.e. from where to where ?
$$(g\cdot \omega)(f(p))(f_*(v_1),\ldots,f_*(v_k)) = g(f(p))\omega(f(p))(f_*(v_1),\ldots,f_*(v_k)) = ((g \circ f) \cdot f^*\omega)(p)(v_1,\ldots,v_k).$$
1) No, a $k$-form is not a vector field. At each point, a $k$-form converts $k$ tangent vectors into a real number.
2) Here $f:\mathbb R^n \to \mathbb R^m$ and $g:\mathbb R^m \to \mathbb R$, in order for $g \cdot \omega$ to make sense.