Find basis for $P_0^\perp \subset P_4$ and $\ker (f \mapsto f(0))$ in $P_4$

Question

Let $P_n \subset \textrm{Map}(\mathbb{C},\mathbb{C})$ the space of polynomial maps $\mathbb{C} \to \mathbb{C}$ with degree $\le n$. We define $\langle f,g\rangle := \int_{-1}^1 f(t)\overline{g(t)}dt$.

i) Find a basis for $P_0^\perp \subset P_4$.

ii) Find a basis of the orthogonal complement of $\ker (f \mapsto f(0))$ in $P_4$.

Answer 1

For first one, you should see, that $P_0$ is generated by $1$ and $P_4$ by $1, x, x^2, x^3, x^4$. So just apply Gramm-Schimdt ortogonalization.

For the second one, you first should identify, what the $\ker$ is.