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LeanPool.JacobianDiffgeo.Finiteness.CompactRestrict

Montel compactness of restrictCLM (finiteness-and-chi) #

Unit: finiteness-and-chi (docs/design/finiteness-and-chi.md §4.4, proof plan §6.3).

Placement note (deviation from the design doc's §3 file plan, flagged honestly): the design's §4.4 header also lists the cocycle-level assembly (isCompactOperator_resZ_UV, isCompactOperator_tradeCompact), but those need ShrinkChain/NZ1/tradeCompact, which are defined in Chain.lean — built after this file per the same file plan's build order. To keep this file gate-free and self-contained (no forward reference to a not-yet-written file), the single-chart lemma lives here; the cocycle-level assembly is proved in Chain.lean immediately after the relevant definitions, by IsCompactOperator.comp_clm/.clm_comp composed with this lemma (per §6.3 step 5) — no mathematical content moves, only the file boundary.

theorem RS.Finiteness.isCompactOperator_of_isCompactOperator_val {E : Type u_2} {N : Type u_3} [NormedAddCommGroup E] [NormedSpace E] [NormedAddCommGroup N] [NormedSpace N] {S : Submodule N} (hS : IsClosed S) (f : E →L[] S) (hf : IsCompactOperator fun (e : E) => (f e)) :

Compat: a CLM into a closed submodule is a compact operator as soon as its composite with the ambient inclusion is (the closedness lets a compact "witness set" for the ambient map be trimmed, via Subtype.isCompact_iff, to a compact witness set inside the submodule itself).

Montel compactness of the restriction: for S' ⋐ S ⊆ source (chartAt ℂ x₀), the restriction BddHoloOn S → BddHoloOn S' is a compact operator.