Theorems · Theorem · global analysis

Bundle.Trivialization.eventually_eq_localFrame_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] [inst_13 : VectorBundle 𝕜 F V] {ι : Type u_7}
  (e : Bundle.Trivialization F Bundle.TotalSpace.proj) [inst_14 : MemTrivializationAtlas e] (b : Module.Basis ι 𝕜 F)
  [inst_15 : ContMDiffVectorBundle 1 F V I] {x : M} {s : (x : M) → V x} [inst_16 : Fintype ι],
  x ∈ e.baseSet →
    ∀ᶠ (x' : M) in nhds x, s x' = ∑ i, (Bundle.Trivialization.localFrameCoeff I e b i x') (s x') • e.localFrame b i x'

A local frame locally spans the space of sections for V: for each local trivialisation e of V around x, we have s = ∑ i, (LinearMap.piApply (b.localFrameCoeff e i) s) • b.localFrame e i near x.

Defined in

Mathlib.Geometry.Manifold.VectorBundle.LocalFrame

Cited by

1 results in Mathlib

Foundations

Depth 222 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceAddCommGroupModuleTopologicalSpaceFiberBundleVectorBundleMemTrivializationAtlasContMDiffVectorBundleFintype