Weak sequential compactness in a Hilbert space (${\sup }_j\Vert u_j{\Vert }_H<\infty ⟹u_{j_k}\rightharpoonup u$)

Every bounded sequence in a Hilbert space has a weakly convergent subsequence. The closed span of a sequence is separable; diagonal convergence of coordinates along an orthonormal basis gives a bounded limiting linear functional, represented by a vector of the Hilbert space. Applied to a bounded family of mollifications, this proves that their strong Lebesgue limit inherits the weak derivative bound.