Theorems · Theorem · algebraic topology

ContinuousOn.clm_bundle_apply

∀ {𝕜 : Type u_8} {B : Type u_9} {F₁ : Type u_10} {F₂ : Type u_11} {M : Type u_13} [inst : NontriviallyNormedField 𝕜]
  {E₁ : B → Type u_14} [inst_1 : (x : B) → AddCommGroup (E₁ x)] [inst_2 : (x : B) → Module 𝕜 (E₁ x)]
  [inst_3 : NormedAddCommGroup F₁] [inst_4 : NormedSpace 𝕜 F₁] [inst_5 : TopologicalSpace (Bundle.TotalSpace F₁ E₁)]
  [inst_6 : (x : B) → TopologicalSpace (E₁ x)] {E₂ : B → Type u_15} [inst_7 : (x : B) → AddCommGroup (E₂ x)]
  [inst_8 : (x : B) → Module 𝕜 (E₂ x)] [inst_9 : NormedAddCommGroup F₂] [inst_10 : NormedSpace 𝕜 F₂]
  [inst_11 : TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [inst_12 : (x : B) → TopologicalSpace (E₂ x)]
  [inst_13 : TopologicalSpace B] [inst_14 : TopologicalSpace M] [inst_15 : FiberBundle F₁ E₁]
  [inst_16 : VectorBundle 𝕜 F₁ E₁] [inst_17 : FiberBundle F₂ E₂] [inst_18 : VectorBundle 𝕜 F₂ E₂] {b : M → B}
  {v : (x : M) → E₁ (b x)} {s : Set M} [inst_19 : ∀ (x : B), IsTopologicalAddGroup (E₂ x)]
  [inst_20 : ∀ (x : B), ContinuousSMul 𝕜 (E₂ x)] {ϕ : (x : M) → E₁ (b x) →L[𝕜] E₂ (b x)},
  ContinuousOn (fun m => ⟨b m, ϕ m⟩) s →
    ContinuousOn (fun m => ⟨b m, v m⟩) s → ContinuousOn (fun m => ⟨b m, (ϕ m) (v m)⟩) s

Consider a C^n map v : M → E₁ to a vector bundle, over a basemap b : M → B, and linear maps ϕ m : E₁ (b m) → E₂ (b m) depending smoothly on m. One can apply ϕ m to v m, and the resulting map is C^n.

Defined in

Mathlib.Topology.VectorBundle.Hom

Cited by

1 results in Mathlib

Foundations

Depth 187 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldAddCommGroupModuleNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceAddCommGroupModuleNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceFiberBundleVectorBundleFiberBundleVectorBundleIsTopologicalAddGroupContinuousSMul