LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)

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Weak closedness keeps the direct-method limit admissible

Statement

Let $X$ be a normed space and let $K \subseteq X$ be weakly sequentially closed (Weak convergence of nets and sequences). If $(u_{j}) \subseteq K$ and $u_{j} ⇀ u$ , then $u \in K$ . In particular, under the Axiom of Choice every nonempty convex norm-closed $K$ has this property, by A norm-closed convex set is weakly sequentially closed.

Facts & Assumptions

Given: A normed space $X$ and a subset $K \subseteq X$ that is weakly sequentially closed: every sequence $(u_{j}) \subseteq K$ with $u_{j} ⇀ u$ satisfies $u \in K$ (Weak convergence of nets and sequences).

[F1]

A set $K$ is weakly sequentially closed when it contains the weak limit of every weakly convergent sequence contained in it; the relation $u_{j} ⇀ u$ denotes convergence in the weak topology $σ (X, X^{*})$ (Weak convergence of nets and sequences).

[F2]

Under the Axiom of Choice, a convex subset of a real or complex normed space that is closed in the norm topology is closed in the weak topology $σ (X, X^{*})$ , hence weakly sequentially closed (A norm-closed convex set is weakly sequentially closed).

Proof

technique · direct, unpacking the definition and quoting the closed-convex lemma

1.1F1given

First assertion. Let $(u_{j}) \subseteq K$ with $u_{j} ⇀ u$ . By the definition [F1] of weak sequential closedness of $K$ recorded in the hypothesis, $u \in K$ ; this is exactly the first sentence of the statement.

2.1F1F2step 1.1∎

Second assertion. Assume additionally that the Axiom of Choice holds and that $K$ is nonempty, convex and norm closed. By [F2] the set $K$ is weakly closed, and a weakly closed set is in particular weakly sequentially closed: if $(u_{j}) \subseteq K$ and $u_{j} ⇀ u$ , then $u$ lies in the weak closure of $K$ , which is $K$ . Hence such a $K$ satisfies the hypothesis of step 1.1 and contains every weak limit of its weakly convergent sequences.

Depends on

Used by

The direct method in a reflexive Banach space Theorem

Dependency tree · two levels

7 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)