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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.

An orthonormal basis of a Hilbert space is an orthonormal set whose linear span has the whole space as its closure. This completeness of an orthonormal system allows every vector to be expanded in the basis vectors, with convergence in norm. In finite dimension it is also a basis in the finite-linear-combination sense.