Skip to content
New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Jump to bottom

[Merged by Bors] - feat: kernel of orthogonalProjection K is Kᗮ #6943

Closed
wants to merge 1 commit into from
Closed

[Merged by Bors] - feat: kernel of orthogonalProjection K is Kᗮ #6943

Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Jump to
Jump to file
Failed to load files.
Diff view
Diff view
11 changes: 11 additions & 0 deletions Mathlib/Analysis/InnerProductSpace/Projection.lean
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -580,6 +580,17 @@ theorem orthogonalProjection_eq_self_iff {v : E} : (orthogonalProjection K v : E
· simp
#align orthogonal_projection_eq_self_iff orthogonalProjection_eq_self_iff

@[simp]
theorem orthogonalProjection_eq_zero_iff {v : E} : orthogonalProjection K v = 0 ↔ v ∈ Kᗮ := by
refine ⟨fun h ↦ ?_, fun h ↦ Subtype.eq <| eq_orthogonalProjection_of_mem_orthogonal
(zero_mem _) ?_⟩
· simpa [h] using sub_orthogonalProjection_mem_orthogonal (K := K) v
· simpa

@[simp]
theorem ker_orthogonalProjection : LinearMap.ker (orthogonalProjection K) = Kᗮ := by
ext; exact orthogonalProjection_eq_zero_iff

theorem LinearIsometry.map_orthogonalProjection {E E' : Type*} [NormedAddCommGroup E]
[NormedAddCommGroup E'] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 E'] (f : E →ₗᵢ[𝕜] E')
(p : Submodule 𝕜 E) [HasOrthogonalProjection p] [HasOrthogonalProjection (p.map f.toLinearMap)]
Expand Down