Reflexivity and weak sequential compactness

For a Banach space X, the canonical map J sends x to the functional on X* that evaluates f at x. Hahn–Banach shows it is an isometry. Reflexivity means J is surjective, not merely that some abstract isomorphism between X and X** happens to exist. Under this canonical identification, weak topology on X corresponds to weak-star topology on X**. Banach–Alaoglu therefore gives weak compactness of the closed unit ball when X is reflexive. The converse also holds, using density of the canonical unit ball in the bidual unit ball and compactness.

Hilbert spaces are reflexive, and so are ℓᵖ spaces for 1<p<∞. The endpoints c₀ and ℓ¹ are not reflexive, despite being familiar complete separable spaces. A closed linear subspace of a reflexive Banach space is reflexive, as is a quotient by a closed subspace. Reflexivity also passes between a Banach space and its dual. These permanence results concern canonical dual structure; they do not say that a reflexive space admits an inner product norm, an orthogonal basis or a bounded projection onto every closed subspace.

For a subset K of a Banach space, weak compactness is equivalent to weak sequential compactness: every sequence in K has a weakly convergent subsequence whose limit remains in K. In the relatively compact version, the subsequential limits may lie in its weak closure. This is the Eberlein–Šmulian theorem. It is a special theorem about weak topology, not a general equivalence for arbitrary compact spaces. In particular, it does not contradict the preceding nonseparable dual-ball example, which used weak-star rather than weak topology.

Combining the theorem with reflexivity shows that every bounded sequence in a reflexive Banach space has a weakly convergent subsequence. No separability assumption on the entire space is necessary. The closed span of a given sequence is separable, which helps explain the countable structure available inside this problem, but the full equivalence is more than an unsupported diagonal argument. The weak limit can have a smaller norm: coordinate unit vectors in ℓ² converge weakly to zero while all have norm one. Thus a weakly compact unit ball need not be norm compact.

A norm-closed convex set is weakly closed by Hahn–Banach separation. In a reflexive Banach space, intersecting such a set with a closed bounded ball yields a weakly compact set. To minimize the norm over a nonempty closed convex set, take a minimizing sequence, which is bounded once one finite competitor is fixed. Extract a weak subsequence, retain feasibility by weak closedness, and use the lower-semicontinuity inequality to obtain a minimizer. Uniqueness requires more geometry, such as strict convexity of the norm, and does not follow from reflexivity alone.

More general direct-method arguments replace the norm by a weakly lower-semicontinuous objective whose sublevel sets supply boundedness. Coercivity is one common sufficient condition for that boundedness. Convex norm-lower-semicontinuous objectives are weakly lower semicontinuous, but an arbitrary norm-continuous nonconvex objective need not have this property. The method therefore separates three tasks: produce a bounded minimizing sequence, keep its weak limit inside the feasible set, and pass the objective inequality to the limit. Reflexivity supplies compactness rather than performing all three tasks by itself.