TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

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The direct method on a weakly closed constraint set

Statement

Assume the ultrafilter lemma, DC and HB (The ultrafilter extension principle (UL/BPI), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, The real dominated-extension principle as an additional hypothesis over ZF). Let $X$ be a real reflexive Banach space (Reflexivity is surjectivity of the canonical map), let $A \subseteq X$ be nonempty and weakly sequentially closed (Weak convergence of nets and sequences), and let $I : X \to (- \infty, + \infty]$ be proper, coercive on $A$ and weakly sequentially lower semicontinuous on $A$ (Proper, coercive and weakly lower semicontinuous extended-real functionals). Then $I$ attains its infimum on $A$ : there is $u_{0} \in A$ with $I (u_{0}) = \inf_{A} I$ . In particular, if $C \subseteq X$ is nonempty, convex and norm closed, and the restriction $I ∣_{C}$ is proper, coercive on $C$ and weakly sequentially lower semicontinuous on $C$ , then $C$ is weakly sequentially closed and the conclusion holds for $A = C$ .

Facts & Assumptions

Given: The ultrafilter lemma, DC and HB; a real reflexive Banach space $X$ ; a nonempty weakly sequentially closed set $A \subseteq X$ ; and a proper functional $I : X \to (- \infty, + \infty]$ that is coercive and weakly sequentially lower semicontinuous on $A$ . For the second assertion, a nonempty convex norm-closed set $C \subseteq X$ such that $I ∣_{C}$ is proper, coercive and weakly sequentially lower semicontinuous on $C$ .

[F1]

The direct method in a reflexive Banach space: under the ultrafilter lemma, DC and HB, for every real reflexive Banach space $X$ , every nonempty weakly sequentially closed $A \subseteq X$ and every proper coercive weakly sequentially lower semicontinuous $I : X \to (- \infty, + \infty]$ , there is $u_{0} \in A$ with $I (u_{0}) = \inf_{A} I$ . In particular, a convex norm-closed set $C$ may be used as the admissible set when $I ∣_{C}$ is proper, coercive and weakly sequentially lower semicontinuous on $C$ .

[F2]

Norm closed convex iff weakly closed: under the assumed HB, every convex norm-closed subset of a normed space is weakly closed and therefore weakly sequentially closed, so every weakly convergent sequence in it has its limit in the set.

[A1]

The ultrafilter extension principle (UL/BPI), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, The real dominated-extension principle as an additional hypothesis over ZF: the three hypotheses recorded in the statement, exactly as consumed by [F1].

Proof

technique · direct

Given: The hypotheses of the statement, including a real reflexive Banach space $X$ and a nonempty weakly sequentially closed $A \subseteq X$ , with $I$ proper, coercive on $A$ and weakly sequentially lower semicontinuous on $A$ .

1.1givenA1F1

All hypotheses of [F1] are met: $X$ is a real reflexive Banach space, $A$ is nonempty and weakly sequentially closed, and $I$ is proper, coercive on $A$ and weakly sequentially lower semicontinuous on $A$ ; the ultrafilter lemma, DC and HB are assumed [A1]. Hence there is $u_{0} \in A$ with $I (u_{0}) = \inf_{A} I$ .

1.2F1F2

For the second assertion let $C \subseteq X$ be nonempty, convex and norm closed, and assume $I ∣_{C}$ is proper, coercive on $C$ and weakly sequentially lower semicontinuous on $C$ . Then $C$ is weakly sequentially closed by [F2], and all hypotheses of [F1] hold with $A = C$ ; hence there is $u_{0} \in C$ with $I (u_{0}) = \inf_{C} I$ .

2.1step 1.1step 1.2A1∎

Step 1.1 proves the first assertion and step 1.2 the "in particular" clause; no convexity or smoothness of a general admissible set $A$ is claimed beyond what is stated, and the three hypotheses of the statement are used only through [F1].

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Sources

Gerald Teschl, Partial Differential Equations: From Classical to Modern (2026 author manuscript; complete 392-page archived text) (standard reference, not scraped)