Theorems · Theorem · functional analysis

Submodule.reflection_mem_subspace_orthogonalComplement_eq_neg

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {K : Submodule 𝕜 E} [inst_3 : K.HasOrthogonalProjection] {v : E}, v ∈ Kᗮ → K.reflection v = -v

The reflection in K of an element of Kᗮ is its negation.

Defined in

Mathlib.Analysis.InnerProductSpace.Projection.Reflection

Cited by

2 results in Mathlib

Foundations

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

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceSubmodule.HasOrthogonalProjection