All the N-tuples of complex numbers $\omega=(w_1,...,w_N)$, all the complex scalars define a inner-product space. The function $\langle \omega, \chi \rangle=\frac{1}N\sum_{i=1}^N w_ix_i^*$ is an inner product for this space.
Let $\lambda_v=2\pi v/N$ and let $[x]$ be the largest integer not exceeding $x$.
It is easy to see that the N vectors:
$z_v=(e^{i\lambda_v},e^{i2\lambda_v},...,e^{iN\lambda_v})$ for $-[(N-1)/2] \leq v \leq [N/2] $
form an orthonormal set, thus a basis for the space.
I don't consider it so straightforward and honestly I don't get how I could conclude, or prove that indeed the vectors are orthonormal.
Any hint or advice will be greatly appreciated.
Just the usual sum of roots of unity proof $$(1-\zeta_N^m)\sum_{j=1}^N\zeta_N^{mj}=\zeta_N^m[1-(\zeta_N^m)^N]=0$$ so if $\zeta_N^m\neq 1$ (i.e., $m\not\equiv 0\pmod{N}$) we have $\sum_{j=1}^N\zeta_N^{mj}=0$, and if $\zeta_N^m=1$ we sum $N$ $1$'s. So $\langle z_v, z_{v'}\rangle=\frac1N\sum_{j=1}^N\zeta_N^{(v-v')j}=\delta_{v',v}$.