Theorems · Theorem · functional analysis
Submodule.mem_orthogonal_singleton_iff_inner_left
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
{u v : E}, v ∈ (𝕜 ∙ u)ᗮ ↔ inner 𝕜 v u = 0A vector in (𝕜 ∙ u)ᗮ is orthogonal to u.
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 165 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- NormedAddCommGroupstatement and proof · cited by 15,752
- Submodulestatement · cited by 7,192
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- Submodule.spanstatement · cited by 1,504
- Inner.innerstatement and proof · cited by 1,089
- Submodule.orthogonalstatement · cited by 257
- inner_eq_zero_symmproof · cited by 12
- Submodule.mem_orthogonal_singleton_iff_inner_rightproof · cited by 10
Cited by7
Results whose statement or proof uses this declaration.
- EuclideanGeometry.Sphere.mem_orthRadius_iff_inner_leftproof · cited by 7
- EuclideanGeometry.inner_vsub_vsub_of_dist_eq_of_dist_eqproof · cited by 3
- Submodule.smul_starProjection_singletonproof · cited by 2
- LinearIsometryEquiv.reflections_generate_dim_auxproof · cited by 1
- InnerProductSpace.gramSchmidt_of_orthogonalproof · cited by 1
- Submodule.reflection_subproof · cited by 1
- stereoInvFunAux_memproof · cited by 1