Let $W'$ be the subspace of $V$ defined by $$W' = \{P(u)(v) + w,\, P \in \mathbb{C}[X] \text{ and } w \in W\}.$$ Verify that $W'$ contains $W$ and $v$ and that it is stable by $u$.
Let $W'$ be the subspace of $V$ defined by
$$W' = \{P(u)(v) + w,\, P \in \mathbb{C}[X] \text{ and } w \in W\}.$$
Verify that $W'$ contains $W$ and $v$ and that it is stable by $u$.