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LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)
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The exact solution set of a weakly closed map is weakly sequentially closed

Statement

Let X,Y be real Hilbert spaces, let D⊆X be nonempty and weakly closed, meaning closed in the coarsest topology making every bounded real-linear functional continuous, and let F:D→Y satisfy this graph condition: whenever xn∈D, xn⇀x, and F(xn)⇀z, one has x∈D and F(x)=z. Weak convergence means convergence under every bounded real-linear functional. For y∈Y, the exact solution set SF(y):={x∈D:F(x)=y} is weakly sequentially closed in X: every sequence in SF(y) that converges weakly to x has x∈SF(y).

Facts & Assumptions

Given: Real Hilbert spaces X,Y, a weakly closed forward map F:D→Y, a datum y∈Y, a sequence (xn) in SF(y), and xn⇀x.

[F1]

Sequential weak-to-weak graph closure means that xn∈D, xn⇀x, and F(xn)⇀z imply x∈D and F(x)=z (Weakly closed nonlinear forward maps on Hilbert spaces).

[F2]

Weak convergence is convergence in the weak topology; in particular, a constant sequence with value y converges weakly to y (Weak convergence of nets and sequences).

Proof

1.1givenF2

Since xn∈SF(y), each xn lies in D and F(xn)=y. Thus (F(xn)) is the constant sequence at y, so it converges weakly to y by [F2].

2.1givenF1step 1.1∎

Apply [F1] to xn⇀x and F(xn)⇀y. It follows that x∈D and F(x)=y, hence x∈SF(y). This is the asserted sequential weak closedness.

Depends on

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Sources