Montel compactness of restrictCLM (finiteness-and-chi) #
Unit: finiteness-and-chi (docs/design/finiteness-and-chi.md §4.4, proof plan §6.3).
isCompactOperator_restrictCLM: forS' ⋐ Sinside a single chart source, the restrictionrestrictCLM : BddHoloOn S →L[ℂ] BddHoloOn S'is a compact operator (Montel's theorem, quoted fromJacobian/Forms/Montel.lean).
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.
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.