Theorems · Theorem · functional analysis
inner_zero_left
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : SeminormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
(x : E), inner 𝕜 0 x = 0- Defined in
- Mathlib.Analysis.InnerProductSpace.Basic
- Cited by
- 59 results in Mathlib
- Foundations
- Depth 49 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- MulZeroClass.zero_mulproof · cited by 1,625
- map_zeroproof · cited by 1,614
- Inner.innerstatement and proof · cited by 1,089
- zero_smulproof · cited by 716
- starRingEndproof · cited by 671
- inner_smul_leftproof · cited by 49
Cited by59
Results whose statement or proof uses this declaration.
- inner_zero_rightproof · cited by 40
- InnerProductGeometry.inner_eq_zero_iff_angle_eq_pi_div_twoproof · cited by 26
- InnerProductGeometry.angle_zero_leftproof · cited by 14
- Submodule.starProjection_eq_self_iffproof · cited by 9
- Submodule.orthogonalProjectionOnto_mem_subspace_eq_selfproof · cited by 8
- InnerProductSpace.gramSchmidt_orthogonalproof · cited by 6
- Submodule.starProjection_singletonproof · cited by 4
- ContinuousLinearMap.ker_adjoint_comp_selfproof · cited by 4
- LinearMap.IsSymmetric.orthogonalFamily_eigenspacesproof · cited by 3
- lp.inner_single_leftproof · cited by 3
- Orientation.eq_zero_or_oangle_eq_iff_inner_eq_zeroproof · cited by 3
- EuclideanGeometry.Sphere.secondInter_zeroproof · cited by 3