If we take a vector space $V\subset \mathbb{P}_n$ (polynomials of degree less or iqual to $n$), I want to prove that if $V$ is stable by derivation then $V=\mathbb{P}_r$ for some $r\leq n$, i.e. if $V=\langle p_1,\dots, p_r\rangle$ and $\langle p_1',\dots, p_r'\rangle \subset \langle p_1,\dots, p_r\rangle$ then $V=\mathbb{P_r}$.
This is my conjecture, perhaps there are other spaces satisfying this condition. Any idea about this?
Use the highest-degree polynomial of V. if f is such a polynomial of grau r, than derivative r times you check that $$a_r \in V$$ implies that $$1 \in V$$, if you derivative r-1 times you check that $$a_{r-1}+a_rx \in V$$ thus $$x \in V$$ analogously you concluded that $$1,x,x^2,...,x^r \in V$$ thus $$V=\mathbb{P}_r$$
Note that $$x^k$$ with $k>r$ is not belongs to $$V$$ for maximality of $$f$$.