Chapter 5.3, Problem 17E

Complete Example 2 by verifying that $\left\{1,x,x^2,x^3\right\}$ is an orthonormal basis for $P_3$ with the inner product $\langle p,q\rangle =a_0{b}_0+a_1{b}_1+a_2{b}_2+a_3{b}_3$.

An Orthonormal basis for $P_3$.

In $P_3$, with the inner product

$\langle p,q\rangle =a_0{b}_0+a_1{b}_1+a_2{b}_2+a_3{b}_3$

The standard basis $B=\left\{1,x,x^2,x^3\right\}$ is orthonormal. The verification of this is left as an exercise (See Exercise 17.).