CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge 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.

The elliptic kernel and cokernel are finite dimensional

Statement

Assume the Axiom of Choice and Countable Choice. In the setting of The Fredholm alternative for weak elliptic Dirichlet problems the weak homogeneous space $N = {u \in H_{0}^{1} (Ω) : a (u, v) = 0 \forall v}$ and the weak adjoint space $N^{*} = {v \in H_{0}^{1} (Ω) : a^{*} (v, w) = 0 \forall w}$ are finite-dimensional over $K$ with $\dim N = \dim N^{*}$ . Explicitly $N = \ker (I - μ K_{μ})$ and $N^{*} = \ker (I - μ K_{μ}^{*})$ as subspaces of $L^{2} (Ω)$ (On bounded domains, the unshifted equation is an identity-minus-compact equation, The adjoint solution operator solves the adjoint form problem), so this common dimension is the dimension of $E_{1} (T) = \ker (T - I)$ (Eigenvalues, eigenvectors, eigenspaces $E_{λ} (T) = \ker (T - λ I)$ , and the spectrum $σ_{F} (T)$ of an endomorphism), for each compact operator $T = μ K_{μ}$ and $T = μ K_{μ}^{*}$ . When the common dimension is positive, $1$ is an eigenvalue and this dimension is its geometric multiplicity; when it is zero, $1$ is not an eigenvalue. No equality of algebraic multiplicities is asserted. The common geometric multiplicity is independent of the admissible shift $μ \geq β$ .

Facts & Assumptions

Given: the Axiom of Choice and Countable Choice; a bounded open set $Ω \subseteq R^{n}$ ; the divergence-form operator with form $a$ and adjoint $a^{*}$ ; a fixed $μ \geq β$ ; the shifted solution operators $K_{μ}, K_{μ}^{*}$ .

[F1]

Identifications: $N = {u \in H_{0}^{1} (Ω) : (I - μ K_{μ}) u = 0}$ and $N^{*} = {v \in H_{0}^{1} (Ω) : (I - μ K_{μ}^{*}) v = 0}$ ; moreover $\ker (I - μ K_{μ}) \subseteq H_{0}^{1} (Ω)$ and $\ker (I - μ K_{μ}^{*}) \subseteq H_{0}^{1} (Ω)$ , so the two sets are exactly the indicated kernels in $L^{2} (Ω)$ (On bounded domains, the unshifted equation is an identity-minus-compact equation, The adjoint solution operator solves the adjoint form problem, The shifted elliptic solution operator, Zero-boundary Sobolev space as a norm closure).

[F2]

Abstract Fredholm dimension: for a compact operator $μ K_{μ}$ on $L^{2} (Ω)$ and $A = I - μ K_{μ}$ , the kernel and the cokernel of $A$ are finite dimensional with equal dimensions; the kernel of the transpose $A^{*}$ is finite dimensional (Fredholm alternative for identity minus compact, Kernel of identity minus compact is finite dimensional, Compact linear operator, The Axiom of Choice). The range of $A = I - μ K_{μ}$ is closed by Range of identity minus compact is closed; AC supplies its DC premise by AC supplies the countable and dependent choices used in Banach integration. By Orthogonal decomposition by a closed subspace, $L^{2} = ran A \oplus (ran A)^{⊥}$ . The adjoint identity gives $(ran A)^{⊥} = \ker (I - μ K_{μ}^{*})$ , and $v \mapsto v + ran A$ restricts to a linear bijection from this orthogonal kernel onto the cokernel.

[F3]

The weak spaces $N, N^{*}$ are the homogeneous and adjoint homogeneous solution spaces of the weak elliptic problem, and the abstract identities of [F1] hold for every admissible shift $μ \geq β$ (The Fredholm alternative for weak elliptic Dirichlet problems).

[F4]

For a linear endomorphism $T$ , its eigenspace for eigenvalue $1$ is $E_{1} (T) = \ker (T - I)$ (Eigenvalues, eigenvectors, eigenspaces $E_{λ} (T) = \ker (T - λ I)$ , and the spectrum $σ_{F} (T)$ of an endomorphism). Its dimension is the geometric multiplicity used here; algebraic multiplicity is not part of this conclusion.

Proof

technique · direct

1.1F1given

Identifications. By [F1], $N = \ker (I - μ K_{μ})$ and $N^{*} = \ker (I - μ K_{μ}^{*})$ as subspaces of $L^{2} (Ω)$ : an element of either kernel lies in $H_{0}^{1} (Ω)$ because the range of the corresponding solution operator is contained in $H_{0}^{1} (Ω)$ .

2.1F2step 1.1given

Finite dimension and equality. Since $μ K_{μ}$ is compact on the Banach space $L^{2} (Ω)$ , [F2] gives $\dim \ker (I - μ K_{μ}) < \infty$ and $\dim \ker (I - μ K_{μ}) = \dim coker (I - μ K_{μ})$ . Under the Riesz identification of $L^{2} (Ω)$ with its dual, the annihilator of the range is $\ker (I - μ K_{μ}^{*})$ , so the cokernel dimension equals $\dim \ker (I - μ K_{μ}^{*})$ ; by step 1.1 these are $\dim N$ and $\dim N^{*}$ , both finite, with $\dim N = \dim N^{*}$ .

3.1F1F3F4step 2.1givenalgebra∎

Independence of the shift and geometric multiplicity. For a further admissible shift $μ^{'} \geq β$ the same argument with $μ^{'}$ gives $N = \ker (I - μ^{'} K_{μ^{'}})$ and $\dim N = \dim N^{*}$ ; the spaces $N, N^{*}$ themselves are defined by the weak equations alone and do not mention any shift, so the common dimension is independent of the choice of admissible $μ$ . By [F4], when positive, these dimensions are the geometric multiplicities of eigenvalue $1$ of $μ K_{μ}$ and $μ K_{μ}^{*}$ , respectively, because $N$ and $N^{*}$ are exactly their eigenspaces. This identifies no algebraic multiplicity.

Depends on

Used by

Uniqueness implies existence for the elliptic Dirichlet problem Corollary

Dependency tree · two levels

81 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

John K. Hunter, Notes on Partial Differential Equations (UC Davis, revised 18 June 2014, complete 242-page notes) (standard reference, not scraped)
Richard S. Laugesen, Linear Analysis and Partial Differential Equations (University of Illinois, 2020, complete 158-page graduate notes) (standard reference, not scraped)