I'm a bit stuck on this question. So if i'm not mistaken in order to prove that $W$ is a subspace of $\Pi_{5}(\Omega)$. I need to show that: $\textit{(i)}$ the zero vector is an element of $W$; $\textit{(ii)}$ $W$ is closed under addition and $\textit{(iii)}$ $W$ is closed under multiplication. (i'm not sure if i'm saying it right, so please correct me if i'm wrong).
$\textbf{(i):}$ $$p(x) = 0\;(\mathrm{zero\;vector}) = 0 + 0x + 0 x^2 + 0x^3 + 0x^4 + 0x^5 = 0$$ $$p'(x) = 0 + 2x0 + 3x^{2}0 + 4x^{3}0 + 5x^{4}0 = 0$$ $$p'(1) = 0$$ $$p'(-1) = 0$$ $$p'(1) + p'(-1) = 0 + 0 = 0$$ so i assume this proves zero vector is an element of $W$.
$\textbf{(ii):}$
Let $p,q \in \Pi_{5}(\Omega)$
I assume that i need to prove $(p+g)'(1) + (p+g)'(-1) = 0$, but how do i do that?
$\textbf{(iii):}$
Let $\lambda \in \mathbb{R}$, and $p \in \Pi_{5}(\Omega)$. I need to prove $(\lambda p)'(-1) + (\lambda p)'(1) = 0$. I also am not sure how to do this.
I would really appreciate your help, thanks in advance!