Fourier sine basis ($e_n(x)=\sqrt{2/\pi }\sin (nx)$)

The functions $\sqrt{2/\pi }\sin (nx)$, $n\ge 1$, form a complete orthonormal basis of $L^2(0,\pi )$. An odd extension to $(-\pi ,\pi )$ reduces completeness to that of the Fourier series. Equality of all sine coefficients therefore implies equality almost everywhere; if both functions are continuous, it implies equality everywhere.

The normalized functions $\sqrt{2/L}\sin (n\pi x/L)$, $n\ge 1$, have completeness of an orthonormal system in L2 space on $(0,L)$. If a function is orthogonal to every sine, extend it oddly to $(-L,L)$. All its Fourier coefficients vanish: the cosine and constant ones by oddness, and the sine ones by assumption. Completeness of the Fourier basis makes the extension zero almost everywhere. This proves the required dense-span property.