A space $\mathcal R$ of integrable functions on the circle with inner product:
$$(f,g)=\frac{1}{2\pi}\int_{0}^{2\pi}f(\theta)\overline {g(\theta)}d\theta$$ and the norm $||f||$ defined by
$$\Vert f\Vert ^{2}=(f,f)=\frac{1}{2\pi}\int_{0}^{2\pi}\vert f(\theta)\vert ^{2}d\theta$$
Lemma 1.5:
Suppose $F$ and $G$ are integrable on the circle with $$F \sim \sum a_{n}e^{in\theta}~~and~~G\sim \sum b_{n}e^{in\theta}$$ Then $$\frac{1}{2\pi}\int_{0}^{2\pi}F(\theta)\overline{G(\theta)}d\theta=\sum_{n=-\infty}^{\infty}a_{n}\overline{b_{n}}.$$
Now I am not sure if the orthonormal property of the family $\{e_{n}(\theta)\}_{n\in \mathbb z}$, that is:
$$(e_{n},e_{m})=\lbrace^{1~~~~if~~n=m}_{0~~~~if~~n\neq m.}$$ Can prove the lemma above or not.
The hint states that from the Parseval's identity:
$$\sum_{n=-\infty}^{\infty}\vert a_{n}\vert ^{2}=\Vert f \Vert ^{2}$$
And
$$(F,G)=\frac{1}{4}[\Vert F+G \Vert ^{2}-\Vert F-G\Vert ^{2}+i(\Vert F+iG\Vert^{2}-\Vert F-iG\Vert^{2})]$$ which holds n every hermitian inner product space.
I don't know how to verify this fact.
Note that the orthonormal realtion is:
$$ \cfrac{1}{2\pi}\int_0^{2\pi}a_je^{ij\theta}e^{-ik\theta} = a_j \delta_{jk}$$
The hint seems unnecessarily complicated , you can simply plug in:
$$F(\theta) = \sum_ja_je^{ij\theta}, \ \ G(\theta)^*\ =\sum_kb^*_ke^{-ik\theta} $$
into the lemma and use the orthonormal relation to get the required result:
$$ \cfrac{1}{2\pi}\int_0^{2\pi}\sum_j\sum_ka_jb_k^*e^{ij\theta}e^{-ik\theta} = \cfrac{1}{2\pi}\sum_j\sum_k\int_0^{2\pi}a_jb_k^*e^{ij\theta}e^{-ik\theta} $$
$$ \hspace{28mm}= \cfrac{1}{2\pi}\sum_{j=k}{2\pi}a_jb_k^*$$
$$ \hspace{17mm}= \sum_{j}a_jb_k^*$$