Theorems · Theorem · global analysis

ContMDiffWithinAt.clm_apply

∀ {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] {F₁ : Type u_8} [inst_6 : NormedAddCommGroup F₁]
  [inst_7 : NormedSpace 𝕜 F₁] {F₂ : Type u_9} [inst_8 : NormedAddCommGroup F₂] [inst_9 : NormedSpace 𝕜 F₂]
  {n : WithTop ℕ∞} {g : M → F₁ →L[𝕜] F₂} {f : M → F₁} {s : Set M} {x : M},
  ContMDiffWithinAt I (modelWithCornersSelf 𝕜 (F₁ →L[𝕜] F₂)) n g s x →
    ContMDiffWithinAt I (modelWithCornersSelf 𝕜 F₁) n f s x →
      ContMDiffWithinAt I (modelWithCornersSelf 𝕜 F₂) n (fun x => (g x) (f x)) s x

Applying a linear map to a vector is smooth within a set. Version in vector spaces. For versions in nontrivial vector bundles, see ContMDiffWithinAt.clm_apply_of_inCoordinates and ContMDiffWithinAt.clm_bundle_apply.

Defined in

Mathlib.Geometry.Manifold.ContMDiff.NormedSpace

Cited by

5 results in Mathlib

Foundations

Depth 200 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace