Question

Let \(\mathcal{P}_{2}\) be the space of polynomials \(p(x)=a+b x+c x^{2}\) of degree at most 2 with the inner product \(\langle p, q\rangle=\int_{-1}^{1} p(x) q(x) d x\). Let \(\ell\) be the functional \(\ell(p):=p(0)\). Find \(h \in \mathcal{P}_{2}\) so that \(\ell(p)=\langle h, p\rangle\) for all \(p \in \mathcal{P}_{2}\).