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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?
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