Theorems · Theorem · functional analysis
Submodule.norm_sq_eq_add_norm_sq_starProjection
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
(x : E) (S : Submodule 𝕜 E) [inst_3 : S.HasOrthogonalProjection],
‖x‖ ^ 2 = ‖S.starProjection x‖ ^ 2 + ‖Sᗮ.starProjection x‖ ^ 2- Cited by
- 3 results in Mathlib
- Foundations
- Depth 184 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Realstatement · cited by 25,697
- RingHom.idstatement · cited by 18,349
- NormedAddCommGroupstatement and proof · cited by 15,752
- Submodulestatement and proof · cited by 7,192
- Norm.normstatement · cited by 5,413
- ContinuousLinearMapstatement · cited by 5,352
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- Submodule.orthogonalstatement · cited by 257
- Submodule.HasOrthogonalProjectionstatement and proof · cited by 245
- Submodule.starProjectionstatement · cited by 92
Cited by3
Results whose statement or proof uses this declaration.
- Submodule.re_inner_starProjection_eq_normSqproof · cited by 1
- Submodule.mem_iff_norm_starProjectionproof · cited by 1
- Submodule.orthogonalProjectionFn_norm_sqproof · cited by 0