Exercise 4

Consider the real vector space of quadratic polynomials $\mathcal{P}_{2}$. Show that this is an inner product operation: $\langle a_{2}x^{2}+a_{1}x+a_{0},b_{2}x^{2}+b_{1}x+b_{0}\rangle=a_{2}b_{2}+a_{1}b_{1}+a_{0}b_{0}$. Consider the resulting norm. What is $\lVert x^{2}-1\rVert$?