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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Bounded Sobolev sequences have strongly convergent subsequences with the weak limit as limit

Statement

Assume the Axiom of Choice. Let $n \geq 2$ , let $Ω \subseteq R^{n}$ be a bounded extension domain, let $1 \leq p < n$ , $p^{*} = \frac{n p}{n - p}$ , and let $(u_{j})$ be bounded in $W^{1, p} (Ω)$ with $u_{j} ⇀ u$ weakly in $W^{1, p} (Ω)$ . Then $u_{j} \to u$ in $L^{q} (Ω)$ for every $1 \leq q < p^{*}$ ; the convergence is of the whole sequence, not merely of a subsequence.

Facts & Assumptions

Given: the Axiom of Choice, a bounded extension domain $Ω \subseteq R^{n}$ , $1 \leq p < n$ , a bounded sequence $(u_{j})$ with $u_{j} ⇀ u$ weakly in $W^{1, p} (Ω)$ , and $1 \leq q < p^{*}$ .

[F1]

Rellich--Kondrachov. Every bounded sequence in $W^{1, p} (Ω)$ has a subsequence converging in $L^{q} (Ω)$ . (The Rellich--Kondrachov theorem for $1 \leq p < n$ on bounded extension domains, Sobolev extension domains and extension operators)

[F2]

Weak convergence tested against $L^{p^{'}}$ . For $g \in L^{p^{'}} (Ω)$ with $1 / p + 1 / p^{'} = 1$ , the functional $v \mapsto \int_{Ω} g v$ is bounded on $W^{1, p} (Ω)$ , so $\int_{Ω} g u_{j} \to \int_{Ω} g u$ . In particular $\int_{A} u_{j} \to \int_{A} u$ for every measurable $A$ of finite measure. For complex-valued classes, these integral identities are read componentwise. (Weak convergence of nets and sequences, Holder's inequality for integrals, including the endpoint cases, The space $L^{p} (μ)$ as the quotient by null functions)

[F3]

Strong convergence tested against $L^{q^{'}}$ . If $v_{k} \to v$ in $L^{q} (Ω)$ and $g \in L^{q^{'}} (Ω)$ , then $\int g v_{k} \to \int g v$ by H"older's inequality. (Holder's inequality for integrals, including the endpoint cases)

Proof

Proof technique: every subsequence has a further $L^{q}$ -convergent subsequence; identify its limit with the weak limit $u$ by testing against finite-measure set indicators; conclude that the whole sequence converges.

1.1F1given

Let $(u_{j_{k}})$ be any subsequence. It is bounded in $W^{1, p} (Ω)$ , so by [F1] it has a further subsequence $(u_{j_{k_{r}}})$ converging in $L^{q} (Ω)$ to some $v$ .

2.1F2F3step 1.1algebra

For every measurable $A \subseteq Ω$ of finite measure, [F2] applied to $g = 1_{A}$ gives $\int_{A} u_{j_{k_{r}}} \to \int_{A} u$ , while [F3] applied to $g = 1_{A} \in L^{q^{'}} (Ω)$ gives $\int_{A} u_{j_{k_{r}}} \to \int_{A} v$ ; hence $\int_{A} (u - v) = 0$ . For each $m \geq 1$ , apply this to the sets where the real or imaginary part of $u - v$ is greater than $1 / m$ or less than $- 1 / m$ , intersected with $B (0, m)$ . Each such set has measure zero, since the corresponding signed part of the integral has magnitude at least its measure divided by $m$ . As $Ω$ is bounded, these sets cover the nonzero real and imaginary parts, so $u = v$ almost everywhere on $Ω$ .

3.1F1step 2.1∎

Thus every subsequence of $(u_{j})$ has a further subsequence converging in $L^{q} (Ω)$ to the same limit $u$ ; in a metric space this forces the whole sequence to converge to $u$ , because otherwise some $ε > 0$ would admit a subsequence staying $ε$ -away from $u$ , and that subsequence would in turn have a further subsequence converging to $u$ . The Axiom of Choice is inherited through [F1], and the weak topology is Hausdorff as recorded in Weak topology is hausdorff.

Remarks

The identification of the strong limit with the weak limit does not use the density of test functions in $L^{q^{'}}$ for $q = 1$ : the finite-measure indicator test functions lie in $L^{p^{'}} \cap L^{q^{'}}$ and separate almost-everywhere classes.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

57 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

Juha Kinnunen, Sobolev Spaces (Aalto University, 2026, complete graduate lecture notes) (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (archived 2025 author manuscript) (standard reference, not scraped)
Richard S. Laugesen, Linear Analysis and Partial Differential Equations (University of Illinois, complete graduate notes) (standard reference, not scraped)