= Weak sequential compactness in a Hilbert space
{title2=$\sup_j\|u_j\|_H<\infty\Longrightarrow u_{j_k}\rightharpoonup u$}
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.
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