Reflexive Banach space

A Banach space $X$ is reflexive when its canonical embedding $J_X:X\to X^{\ast \ast }$ is surjective. Equivalently, its closed unit ball is weakly compact.

A nonempty norm-closed convex subset $C$ of a reflexive Banach space attains its distance to zero. For $d={\inf }_C\Vert x\Vert$, the sets $C\cap (d+1/n)B_X$ are nonempty weakly closed subsets of one weakly compact ball, and form a decreasing family. Compactness gives a point in their intersection, of norm $d$. Mazur theorem supplies weak closedness of $C$ and the balls; no sequential-compactness theorem is required.

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.