I would like to evaluate the inner product of two components of a basis function that is part of a complete set. The two components are the imaginary and real components of the basis function. Could I expect my answer to be zero? I have tried to consider them both as imaginary and take their Hermitian inner product but that hasn't produced anything interesting (and nor did I expect it to).
\int_a^b r(x)t(x) dx = 0
I gather we're talking about $L^2(\mu)$ for some measure $\mu$. Any function of norm one is part of a complete orthonomal set, so no you certainly cannot expect the inner product of the real and imaginary parts to vanish.