Polynomial subspace Wondering abut this set : $E=(p(X) \ \in \mathbb{R}[X]; Xp(X)+p'(X)=0)$, is it a subspace of $\mathbb{R}[X] $? I definitely think it is because it only includes the zero polynomial but how could we prove it ? I usually take $u$ and $v$ which are in the set and then prove that $\lambda u+v \ \in$ the set but don't see how to proceed here.

If you already see that the set only includes the zero polynomial, then all you need to do is, as always, prove that if $u,v\in E$, then so is $\lambda u +v \in E$. There's nothing special that requires you to change the definition you're using - however, if $u,v\in E$ and $E$ contains only the zero polynomial, then $u=v= 0$ meaning $\lambda u + v=0$ and since $0\in E$, we get $\lambda u + v \in E$ for any choice of $u$, $v$, or $\lambda$.