The trigonometric characters are orthonormal in $L^2$ of the torus

$e_k\overline{e_l}=e_{k-l}$ and $\mathrm{\mid }{e}_k\mathrm{\mid }=1$; in particular the function $t\mapsto e_m(q(t))$ on $\mathbb{R}$ equals $\exp (2\pi imt)=\cos (2\pi mt)+i\sin (2\pi mt)$, and it has period $1$ (Fourier coefficients and trigonometric polynomials on the torus, $\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$).

For a continuous $1$-periodic $g:\mathbb{R}\to \mathbb{C}$ the torus integral equals the Riemann integral over $[0,1]$: ${\int }_{\mathbb{T}}Gd{m}_{\mathbb{T}}={\int }_0^{1}g(t)dt$ for $G(q(t))=g(t)$, because the torus integral is represented on $[0,1)$ and a bounded Riemann integrable function on $[0,1]$ is Lebesgue measurable with the same integral, the endpoint being a null set (The one-dimensional torus and its normalized Haar integral, A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral).

A continuous real function on a closed interval is Riemann integrable, and for differentiable $G$ with $G^{\mathrm{\prime }}=g$ integrable, ${\int }_a^{b}g=G(b)-G(a)$; the derivatives of sine and cosine are cosine and minus sine, and the chain rule computes the derivatives of $t\mapsto \sin (2\pi mt)$ and $t\mapsto \cos (2\pi mt)$ (A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$, The derivatives of sine and cosine are cosine and minus sine, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$).

$\sin (m\pi )=0$ and $\cos (m\pi )=(-1{)}^m$ for every integer $m$: the zero-set theorem gives the sine values, while the shift formula $\cos (x+\pi )=-\cos x$ and $\cos 0=1$ give the cosine values by integer induction (The zero sets of sine and cosine and the least positive common period 2 pi, Quarter-turn values and shifts by pi/2 and pi).

The pairing on complex $L^2$ is linear in the first variable and conjugate-linear in the second, with $⟨f,g⟩=\int f\bar{g}$, and the Fourier coefficient is $\widehat{f}(k)=⟨f,e_k⟩$ ($L^2$ with the integral pairing is a Hilbert space, Fourier coefficients and trigonometric polynomials on the torus).