= Completeness of the Fourier sine basis
The normalized functions $\sqrt{2/L}\sin(n\pi x/L)$, $n\geq1$, 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.
Back to article page
