Domain of Sturm-Liouville operators

The relevant Hilbert space is not the subspace of [itex]C^2[/itex] itself (which is not complete with respect to the [itex]L^2[/itex] norm anyway), but the completion of that subspace with respect to the [itex]L^2[/itex] norm.

 

As the set of equivalence classes of square-integrable but not necessarily continuous functions satisfying the given boundary conditions under the equivalence relation [itex]{\sim}[/itex], where [itex]f \sim g[/itex] if and only if the set [itex]\{ x : f(x) \neq g(x)\}[/itex] has measure zero. This construction is necessary to ensure that [itex]\left(\int f^2\,\mathrm{d}x\right)^{1/2}[/itex] remains a norm, since two functions which differ on a set of measure zero have the same integral. It also ensures that no two continuous functions are in the same equivalence class, since if two continuous functions differ at a point then they must differ on an open neighbourhood of that point, and open neighbourhoods have strictly positive measure.