Exercise 2 Define on $\mathbb{R}_{2\pi}$ a mapping
$\langle \cdot, \cdot \rangle : \mathbb{R}_{2\pi} \times \mathbb{R}_{2\pi} \to \mathbb{C}$, $\langle f, g \rangle := \frac{1}{2\pi} \int_0^{2\pi} f(x)\overline{g(x)}dx$.
Prove that, for $f, g, h \in \mathbb{R}_{2\pi}$ and $\lambda \in \mathbb{C}$, the following properties hold:
(a) $\langle f + g, h \rangle = \langle f, h \rangle + \langle g, h \rangle$.
(b) $\langle f, g + h \rangle = \langle f, g \rangle + \langle g, h \rangle$.
(c) $\langle \lambda f, g \rangle = \lambda \langle f, g \rangle$ and $\langle f, \lambda g \rangle = \overline{\lambda} \langle f, g \rangle$.
(d) $\langle f, g \rangle = \overline{\langle g, f \rangle}$.
