"Theorem": Consider two spaces $W$ and $U$ of finite dimension. If $W \subset U$ then $W^*=U^*$.
Proof:
$W^\ast$ is a subset of $U^\ast$: $ \begin{align*} W &= \langle e_1, \ldots, e_k \rangle, \\ U& = \langle e_1, \ldots, e_k, \ldots, e_n \rangle, \end{align*}$ then $ \begin{align*} W^\ast &= \langle e^\ast_1, \ldots, e^\ast_k \rangle, \\ U^\ast &= \langle e^\ast_1, \ldots, e^\ast_k, \ldots, e^\ast_n \rangle. \end{align*}$ It's clear to see that $W^* \subset U^*$.
$U^\ast$ is a subset of $W^\ast$:
Any functional $U^\ast \ni f: U \to \bf R$ is also functional over $W$, so $U^\ast \subset W^\ast$.
More exactly: $ f(U) \subset {\bf R} \Rightarrow f (W \subset U) \subset {\bf R}.$
So $W^\ast = U^\ast$.
Where is mistake?