If $f$ and $g$ are complex-valued continuous Z-periodic functions, then by definition, $\|f + g\|_2^2 - \|f - g\|_2^2 = \int_0^{1}((f + g)(\overline{f + g}) - (f - g)(\overline{f - g})) = \int_0^{1}2(f\overline{g} + \overline{f}g) = 4\Re\int_0^{1}f\overline{g} = 4\Re\langle f,g\rangle = 2\langle f + g,f + g\rangle.$
On the other hand, it also seems that $\|f + g\|_2^2 - \|f - g\|_2^2 = \langle f + g,f + g\rangle - \langle f - g,f - g\rangle = \langle f + g,f + g\rangle + \langle f - g,-(f - g)\rangle = 2\langle f,g \rangle.$
What is this disparity and where exactly do I misunderstand?