Theorems · Theorem · functional analysis
Submodule.orthogonalProjectionFn_norm_sq
Deprecated since 2026-06-10Use Submodule.norm_sq_eq_add_norm_sq_starProjection instead.
∀ {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‖ * ‖v‖ =
‖v - Submodule.orthogonalProjectionFn v‖ * ‖v - Submodule.orthogonalProjectionFn v‖ +
‖Submodule.orthogonalProjectionFn v‖ * ‖Submodule.orthogonalProjectionFn v‖- Cited by
- 0 results in Mathlib
- Foundations
- Depth 185 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Realstatement · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- Submodulestatement and proof · cited by 7,192
- Norm.normstatement and proof · cited by 5,413
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- add_commproof · cited by 1,535
- sqproof · cited by 280
- Submodule.HasOrthogonalProjectionstatement and proof · cited by 245
- Submodule.starProjectionproof · cited by 92
- Submodule.starProjection_orthogonal_valproof · cited by 7
Cited by0
Results whose statement or proof uses this declaration.
Nothing cites this yet.