I'm confused about this question and I can't find anything helpful anywhere...
My question is as follows:
A set of functions $$ \phi_0,\phi_1, \phi_2, ..., \phi_n $$ is said to be linearly independent if any linear combination of the form $$ a_0\phi_0+a_1\phi_1+a_2\phi_2+...+\phi_n=0 $$ imply that the numbers $a_0, ... a_n$ are all zero for any chosen $n \geq 1$. Now suppose that $f_0, f_1, f_2, ...$ is a linearly independent system in $C_p(a,b)$. Define a new system of functions $e_0, e_1, e_2, ...$ as follows: $$ g_0=f_0 \phantom{1333331133333333333333333} e_0=\frac{g_0}{||g_0||} $$ $$ g_1=f_1- \frac{\langle f_1, g_0\rangle}{||g_0||^2}g_0 \phantom{11111111111111}e_1=\frac{g_1}{||g_1||} $$ $$ g_2 = f_2 - \frac{\langle f_2, g_1\rangle}{||g_1||^2}g_1 -\frac{\langle f_2, g_0\rangle}{||g_0||^2}g_0 \phantom{111} e_2=\frac{g_2}{||g_2||} $$ show that the new system $e_0, e_1, e_2, ... $ is an orthonormal set in $C_p(a, b)$
My confusion lies in how you show something is an orthonormal set. I get I need to use induction but yeah that's as much as I understand needs to be done. Any help is appreciated.