\|f - g\| = \sqrt{\int_{-\pi}^{\pi} \|f(x) - g(x)\|^2 dx}
where $f : [-\pi, \pi] \to \mathbb{R}$ and $g : [-\pi, \pi] \to \mathbb{R}$ are given functions for which the integral makes sense.
Note: While we can think of $\|f - g\|$ as a \"distance\" between $f$ and $g$ and even think of $\|f\| = \|f - 0\|$ as a \"length\" of $f$, it is not the usual absolute value for real numbers. However, $|f(x) - g(x)|$ inside the integral is the standard absolute value.
Let $f_n(x) = \pi^{-1/2} \sin(nx)$. Calculate $\|f_n\|^2$, $\|f_m\|^2$, and $\|f_n - f_m\|^2$ for $n, m \ge 1$.
