Theorems · Theorem · global analysis

IsLocalFrameOn.eventually_eq_sum_coeff_smul

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {H : Type u_3} [inst_3 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4}
  [inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] {F : Type u_5} [inst_6 : NormedAddCommGroup F]
  [inst_7 : NormedSpace 𝕜 F] {V : M → Type u_6} [inst_8 : TopologicalSpace (Bundle.TotalSpace F V)]
  [inst_9 : (x : M) → AddCommGroup (V x)] [inst_10 : (x : M) → Module 𝕜 (V x)]
  [inst_11 : (x : M) → TopologicalSpace (V x)] [inst_12 : FiberBundle F V] {ι : Type u_7} {s : ι → (x : M) → V x}
  {u : Set M} {n : WithTop ℕ∞} {x : M} [inst_13 : Fintype ι] (hs : IsLocalFrameOn I F n s u) (t : (x : M) → V x),
  u ∈ nhds x → ∀ᶠ (x' : M) in nhds x, t x' = ∑ i, (hs.coeff i x') (t x') • s i x'

A local frame locally spans the space of sections for V: for each local frame s i on an open set u around x, we have t = ∑ i, hs.coeff i x (t x) • (s i x) near x.

Defined in

Mathlib.Geometry.Manifold.VectorBundle.LocalFrame

Cited by

2 results in Mathlib

Foundations

Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceAddCommGroupModuleTopologicalSpaceFiberBundleFintype