Let \(\mathcal{P}_{2}\) be the space of quadratic polynomials.
a) Show that \(\langle f, g\rangle=f(-1) g(-1)+f(0) g(0)+f(1) g(1)\) is an inner product for this space.
b) Using this inner product, find an orthonormal basis for \(\mathcal{P}_{2}\).
c) Is this also an inner product for the space \(\mathcal{P}_{3}\) of polynomials of degree at most three? Why?