Exercise 23

Let $\mathcal{P}_{j}(\mathbb{R})$ be the vector space over the reals of degree $j$ polynomials. Show that if $j\leq k$ then $\mathcal{P}_{j}(\mathbb{R})$ is an invariant subspace of $\mathcal{P}_{k}(\mathbb{R})$ under the differentiation operator. In $\mathcal{P}_{7}(\mathbb{R})$, does any of $\mathcal{P}_{0}(\mathbb{R})$, …, $\mathcal{P}_{6}(\mathbb{R})$ have an invariant complement?