My question is regarding Axler's proof of this Algebraic Property of dual maps: $(ST)' = T'S'$ for $T\in\mathcal{L}(U,V), S\in\mathcal{L}(V,W)$ where $T',S'$ represents the dual map.
In the last step of the proof, he says that $T'(S'(\varphi)) = (T'S')(\varphi)$ by the definition of composition.
I understand composition to be $(f\circ g)(x) = f(g(x))$, where $f,g$ are functions. Which I thought was different from $(fg)(x)$. Am I misunderstanding that? Is multiplying functions together really the same as the composition? Or is there some property of dual maps that make this possible?
Source: Linear Algebra Done Right by Sheldon Axler, 3rd ed.
What you're saying is correct: In the case of linear maps multiplication is precisely composition. If you want some intution on why that's the case think about the fact that a linear map can be represented by a matrix.