Theorems · Theorem · functional analysis
Submodule.orthogonalProjection_mem_subspace_eq_self
Deprecated since 2026-05-05Use Submodule.orthogonalProjectionOnto_mem_subspace_eq_self 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 : ↥K), K.orthogonalProjectionOnto ↑v = vAlias of Submodule.orthogonalProjectionOnto_mem_subspace_eq_self.
The orthogonal projection sends elements of K to themselves.
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- Foundations
- Depth 180 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites10
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- DFunLike.coestatement · cited by 62,936
- RingHom.idstatement · cited by 18,349
- NormedAddCommGroupstatement · cited by 15,752
- Submodulestatement · cited by 7,192
- ContinuousLinearMapstatement · cited by 5,352
- InnerProductSpacestatement · cited by 3,523
- RCLikestatement · cited by 2,829
- Submodule.HasOrthogonalProjectionstatement · cited by 245
- Submodule.orthogonalProjectionOntostatement · cited by 103
- Submodule.orthogonalProjectionOnto_mem_subspace_eq_selfproof · cited by 8
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