Theorems · Definition · algebraic topology

Bundle.Pretrivialization.continuousLinearMap

{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)] →
                                                [FiberBundle F₁ E₁] →
                                                  [inst_15 : (x : B) → TopologicalSpace (E₂ x)] →
                                                    [FiberBundle F₂ E₂] →
                                                      [Bundle.Trivialization.IsLinear 𝕜₁ e₁] →
                                                        [Bundle.Trivialization.IsLinear 𝕜₂ e₂] →
                                                          Bundle.Pretrivialization (F₁ →SL[σ] F₂) Bundle.TotalSpace.proj

Given trivializations e₁, e₂ for vector bundles E₁, E₂ over a base B, Pretrivialization.continuousLinearMap σ e₁ e₂ is the induced pretrivialization for the continuous σ-semilinear maps from E₁ to E₂. That is, the map which will later become a trivialization, after the bundle of continuous semilinear maps is equipped with the right topological vector bundle structure.

Defined in

Mathlib.Topology.VectorBundle.Hom

Cited by

5 results in Mathlib

Foundations

Depth 165 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNontriviallyNormedFieldNormedAddCommGroupNormedSpaceAddCommGroupModuleTopologicalSpaceNormedAddCommGroupNormedSpaceAddCommGroupModuleTopologicalSpaceTopologicalSpaceTopologicalSpaceFiberBundleTopologicalSpaceFiberBundleBundle.Trivialization.IsLinearBundle.Trivialization.IsLinear