I am reading Clifford Algebra: An Introduction, by London Mathematical Society. On page 15, I have encountered the statement, which I rephrase as follows. Let $T$ be a linear mapping from vector space $E$ to $F$. If $\varphi \in F'$ and $x \in E$, let $(T'(\varphi))(x) = \varphi(T(x))$. Then, $T(\varphi) \in E'$, and $T'$ is a linear mapping of $F'$ into $E'$; it is the transposed mapping of $T$.
I can understand the whole statement, but am only not sure whether the phrase "Then, $T(\varphi) \in E'$" is a typo of the book. In my mind, the correct phrase should have been "Then, $T'(\varphi) \in E'$". By definition, $T(x) \in F$. If not a typo of the book, I can not see why $T(\varphi) \in E'$.
Any help is appreciated.