I watched a video recently which used the formula $$C_k=\frac{\int f_1(t)f_2(t)dt}{\int f_2(t)^2dt}$$ given that
$$f_1(t) = C_k f_2(t)$$
$C_k$ is a coefficent that can convert any real function into another real function, which is incredibly useful. However I could not figure out how it was derived. What is the proof for this?
Here is the video that I am referring to.