Theorems · Theorem · algebraic topology

Bundle.Pretrivialization.continuousLinearMap_symm_apply

∀ {𝕜₁ : Type u_1} [inst : NontriviallyNormedField 𝕜₁] {𝕜₂ : Type u_2} [inst_1 : NontriviallyNormedField 𝕜₂]
  (σ : 𝕜₁ →+* 𝕜₂) {B : Type u_3} {F₁ : Type u_4} [inst_2 : NormedAddCommGroup F₁] [inst_3 : NormedSpace 𝕜₁ F₁]
  {E₁ : B → Type u_5} [inst_4 : (x : B) → AddCommGroup (E₁ x)] [inst_5 : (x : B) → Module 𝕜₁ (E₁ x)]
  [inst_6 : TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_6} [inst_7 : NormedAddCommGroup F₂]
  [inst_8 : NormedSpace 𝕜₂ F₂] {E₂ : B → Type u_7} [inst_9 : (x : B) → AddCommGroup (E₂ x)]
  [inst_10 : (x : B) → Module 𝕜₂ (E₂ x)] [inst_11 : TopologicalSpace (Bundle.TotalSpace F₂ E₂)]
  [inst_12 : TopologicalSpace B] (e₁ : Bundle.Trivialization F₁ Bundle.TotalSpace.proj)
  (e₂ : Bundle.Trivialization F₂ Bundle.TotalSpace.proj) [inst_13 : (x : B) → TopologicalSpace (E₁ x)]
  [inst_14 : FiberBundle F₁ E₁] [inst_15 : (x : B) → TopologicalSpace (E₂ x)] [inst_16 : FiberBundle F₂ E₂]
  [inst_17 : Bundle.Trivialization.IsLinear 𝕜₁ e₁] [inst_18 : Bundle.Trivialization.IsLinear 𝕜₂ e₂]
  (p : B × (F₁ →SL[σ] F₂)),
  ↑(Bundle.Pretrivialization.continuousLinearMap σ e₁ e₂).symm p =
    ⟨p.1, Bundle.Trivialization.symmL 𝕜₂ e₂ p.1 ∘SL p.2 ∘SL Bundle.Trivialization.continuousLinearMapAt 𝕜₁ e₁ p.1⟩

Defined in

Mathlib.Topology.VectorBundle.Hom

Cited by

0 results in Mathlib

Foundations

Depth 166 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNontriviallyNormedFieldNormedAddCommGroupNormedSpaceAddCommGroupModuleTopologicalSpaceNormedAddCommGroupNormedSpaceAddCommGroupModuleTopologicalSpaceTopologicalSpaceTopologicalSpaceFiberBundleTopologicalSpaceFiberBundleBundle.Trivialization.IsLinearBundle.Trivialization.IsLinear