Theorems · Theorem · functional analysis
Submodule.norm_orthogonalProjection_apply_le
Deprecated since 2026-05-05Use Submodule.norm_orthogonalProjectionOnto_apply_le 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), ‖K.orthogonalProjectionOnto v‖ ≤ ‖v‖Alias of Submodule.norm_orthogonalProjectionOnto_apply_le.
The orthogonal projection onto a closed subspace is norm non-increasing.
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- Foundations
- Depth 178 from the axioms · uses propext, Classical.choice, Quot.sound
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- DFunLike.coestatement · cited by 62,936
- Realstatement · cited by 25,697
- RingHom.idstatement · cited by 18,349
- NormedAddCommGroupstatement · cited by 15,752
- Submodulestatement · cited by 7,192
- Norm.normstatement · cited by 5,413
- 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.norm_orthogonalProjectionOnto_apply_leproof · cited by 2
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