I am trying to be advance in linear algebra but I dont have enough resources. I tried to solve this question but I couldn't.
Let V be the real inner product space of the real valued functions on the interval $−1≤t≤1$ with inner product $$ <f\mid g>= \int_0^1 f(x)g(x)\,dx. $$
Suppose that $f_n(x)=\sqrt2{\cos|2πnx|}$ and $g_n(x)=\sqrt2{\sin|2πnx|}$. Show that $(1,f_1,g_1,f_2,g_2,…)$ is a infinite orthonormal set.