A Banach space is reflexive if and only if its closed unit ball is compact in the weak topology. One direction applies the Banach-Alaoglu theorem to the unit ball in the bidual; the converse follows because Goldstine theorem makes the canonical image of the unit ball weak-star dense, while weak compactness makes that image weak-star closed.
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.
Every bounded sequence in a reflexive Banach space has a weakly convergent subsequence. This is the sequential form of weak compactness supplied by the Eberlein-Šmulian theorem.
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