Why is the math in the red box necessary? According to this definition, it isn't:
sorry, i don't understand your question …
the red box proves that (φ0, φn) = 0 (for n ≠ 0)
According to definition 12.1.3, a set of real-valued functions can be proven to be orthogonal if (φm, φn) = 0. So why is it necessary to prove (φ0, φn) = 0?
m=0 is contained as a particular case for arbitrary m and n. It's no need to make the particular case. The proof goes directly by putting cos a = Re (e^ia).
So you're saying it was unnecessary?
That went over my head.
because φo is a member of the set
That's exactly what I meant.
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