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A bounded sequence in a reflexive Banach space has a weakly convergent subsequence

Statement

Assume the ultrafilter lemma, DC and HB. Let $X$ be a real reflexive Banach space (Reflexivity is surjectivity of the canonical map) and let $(u_{j})$ be a norm-bounded sequence in $X$ . Then $(u_{j})$ has a subsequence converging weakly to a point of $X$ (Weak convergence of nets and sequences).

Facts & Assumptions

Given: The ultrafilter lemma (The ultrafilter extension principle (UL/BPI)), the principle of dependent choice (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain) and HB (The real dominated-extension principle as an additional hypothesis over ZF); a real reflexive Banach space $X$ (Reflexivity is surjectivity of the canonical map); and a norm-bounded sequence $(u_{j}) \subseteq X$ .

[F1]

Under the ultrafilter lemma, DC and HB, a real Banach space $X$ is reflexive if and only if every norm-bounded sequence in $X$ has a subsequence that converges weakly to a point of $X$ (Reflexivity is equivalent to weak subsequential compactness of bounded sequences); the convergence is in the sense of Weak convergence of nets and sequences.

Proof

technique · direct, by the forward implication of the reflexivity characterisation

1.1F1given

The three choice principles named in the hypothesis are exactly the ones assumed by [F1], and $X$ is a real reflexive Banach space by hypothesis; the sequence $(u_{j})$ is norm bounded by hypothesis. The forward implication of [F1] therefore applies and produces a strictly increasing sequence of indices $j_{1} < j_{2} < \dots$ and a point $u \in X$ with $u_{j_{k}} ⇀ u$ .

2.1step 1.1∎

The limit $u$ obtained in step 1.1 is a point of $X$ , so $(u_{j})$ has a subsequence converging weakly to a point of $X$ , which is the stated conclusion.

Depends on

Used by

A minimising sequence need not converge strongly Counterexample
Higher eigenvalues by orthogonality-constrained minimisation Theorem
The direct method in a reflexive Banach space Theorem
The first Dirichlet eigenfunction by constrained minimisation Theorem

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gerald Teschl, Partial Differential Equations: From Classical to Modern (2026 author manuscript; complete 392-page archived text) (standard reference, not scraped)
Francesco Paolo Maiale (course by Giovanni Alberti), Lecture Notes Calculus of Variations A, University of Pisa (last update 21 August 2019; complete 149-page notes) (standard reference, not scraped)