Let $n$ be a positive integer, and let $V$ be a $\mathbb{C}$-vector space of complex-valued functions on $\mathbb{R}$, generated by the set $\{\cos k x, \sin k x ; k=0,1, \ldots, n-1\}$.

(i) Let $\langle f, g\rangle=\int_{0}^{2 \pi} f(x) \overline{g(x)} d x$ for $f, g \in V$. Show that this is a positive definite Hermitian form on $V$.

(ii) Let $\Delta(f)=\frac{d^{2}}{d x^{2}} f(x)$. Show that $\Delta$ is a self-adjoint linear transformation of $V$ with respect to the form defined in (i).

(iii) Find an orthonormal basis of $V$ with respect to the form defined in (i), which consists of eigenvectors of $\Delta$.