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On pg 10, the example says f(x)=/=0 while R.H.S is zero. It is an equations started from the assumption in pg 9; f(x)=c0f(x)0+c1f(x)1…, then how do we get inequality?

if the system is complete and orthogonal, then (f(x),ϕ_n(x))=0, which makes sense only when f(x)=0.

but we know for fourier series, we get values for Rhs and Lhs.