Chapter 5 - Inner Product Spaces - 5.2 Inner Product Spaces - 5.2 Exercises - Page 246: 51

$\theta=\frac{\pi}{2}$.

Let $f(x)=x, \quad g(x)=x^2$, then $\langle f, g\rangle=\int_{-1}^{1}f(x)g(x) d x=\int_{-1}^{1} x^{3}d x=\left[\frac{1}{4}x^{4}\right]_{-1}^{1}=0$ The angle $\theta$ between $f$ and $g$ is given by the formula $$\cos \theta=\frac{\langle f, g\rangle}{\|f\| \cdot\|g\|}=\frac{0}{\|f\| \cdot\|g\|}=0.$$ That is $\theta=\frac{\pi}{2}$.