Theorems · Theorem · functional analysis

Submodule.reflection_map_apply

∀ {𝕜 : Type u_1} [inst : RCLike 𝕜] {E : Type u_4} {E' : Type u_5} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedAddCommGroup E'] [inst_3 : InnerProductSpace 𝕜 E] [inst_4 : InnerProductSpace 𝕜 E'] (f : E ≃ₗᵢ[𝕜] E')
  (K : Submodule 𝕜 E) [inst_5 : K.HasOrthogonalProjection] (x : E'),
  (Submodule.map (↑f.toLinearEquiv) K).reflection x = f (K.reflection (f.symm x))

Reflection in the Submodule.map of a subspace.

Defined in

Mathlib.Analysis.InnerProductSpace.Projection.Reflection

Cited by

1 results in Mathlib

Foundations

Depth 184 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupNormedAddCommGroupInnerProductSpaceInnerProductSpaceSubmodule.HasOrthogonalProjection