Consider the definition of dual map, $T'$ in Axler (2015).
If $T\in\mathcal{L}(V,W)$, then the dual map of $T$ is the linear map $T'\in\mathcal{L}(W',V')$ defined by $T'(\varphi)=\varphi\circ T$ for $\varphi\in W'$.
My Question:
(1) I don't understand how the composition $\varphi\circ T$ works. $T$ maps from $V$ to $W$. Then, how does $\varphi\circ T$ gives you a linear map from $V$ to $\mathcal{F}$ since $\varphi$ is an element of $W'$, how does this composition works?
Reference:
Axler, Sheldon J. $\textit{Linear Algebra Done Right}$, New York: Springer, 2015.
Best Answer
$W'$ is the dual space of $W$: i.e. it is shorthand for $\mathcal{L}(W,\mathcal{F})$.
Thus $\varphi \in W'$ is a map $W \rightarrow \mathcal{F}$, and $\varphi \circ T$ is a map $V \rightarrow W \rightarrow \mathcal{F}$, which means it is an element of $\mathcal{L}(V,\mathcal{F})$ aka $V'$.
Thus $T'$, which takes an element $\varphi \in W'$ and returns $\varphi \circ T \in V'$, is an element of $\mathcal{L}(W',V')$