Theorems · Definition · algebraic topology

Bundle.Trivialization.continuousLinearEquivAt

(R : Type u_1) →
  {B : Type u_2} →
    {F : Type u_3} →
      {E : B → Type u_4} →
        [inst : NontriviallyNormedField R] →
          [inst_1 : (x : B) → AddCommMonoid (E x)] →
            [inst_2 : (x : B) → Module R (E x)] →
              [inst_3 : NormedAddCommGroup F] →
                [inst_4 : NormedSpace R F] →
                  [inst_5 : TopologicalSpace B] →
                    [inst_6 : TopologicalSpace (Bundle.TotalSpace F E)] →
                      [inst_7 : (x : B) → TopologicalSpace (E x)] →
                        [FiberBundle F E] →
                          (e : Bundle.Trivialization F Bundle.TotalSpace.proj) →
                            [Bundle.Trivialization.IsLinear R e] → (b : B) → b ∈ e.baseSet → E b ≃L[R] F

In a vector bundle, a trivialization in the fiber (which is a priori only linear) is in fact a continuous linear equiv between the fibers and the model fiber.

Defined in

Mathlib.Topology.VectorBundle.Basic

Cited by

27 results in Mathlib

Foundations

Depth 79 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldAddCommMonoidModuleNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceFiberBundleBundle.Trivialization.IsLinear