= Solution
The <dual space> is the <vector space> of all <linear functionals> on $V$:
$$
\boxed{V'=\operatorname{Hom}_{\mathbb C}(V,\mathbb C)}.
$$
Addition and <scalar multiplication> are defined pointwise. The <dual map> sends an output measurement back along $T$:
$$
\boxed{T':V'\to V',\qquad T'(\varphi)=\varphi\circ T,\qquad (T'\varphi)(v)=\varphi(Tv)}.
$$
The <function composition> of these <linear maps> is linear, and $T'$ is linear in $\varphi$. This is the algebraic <dual map>, so its definition involves no <complex conjugation> or <inner product>.
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