Theorems · Theorem · functional analysis
Submodule.mem_orthogonal
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
(K : Submodule 𝕜 E) (v : E), v ∈ Kᗮ ↔ ∀ u ∈ K, inner 𝕜 u v = 0When a vector is in Kᗮ.
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 162 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- NormedAddCommGroupstatement and proof · cited by 15,752
- Submodulestatement and proof · cited by 7,192
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- Inner.innerstatement · cited by 1,089
- Submodule.orthogonalstatement · cited by 257
Cited by6
Results whose statement or proof uses this declaration.
- Submodule.inner_right_of_mem_orthogonalproof · cited by 8
- Submodule.top_orthogonal_eq_botproof · cited by 6
- EuclideanGeometry.orthogonalProjection_mapproof · cited by 3
- Submodule.orthogonal_eq_interproof · cited by 1
- Submodule.norm_projection_orthogonal_leproof · cited by 1
- Submodule.norm_sq_eq_add_norm_sq_projectionproof · cited by 1