Theorems · Theorem · functional analysis
inner_self_eq_norm_sq_to_K
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : SeminormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
(x : E), inner 𝕜 x x = ↑‖x‖ ^ 2- Defined in
- Mathlib.Analysis.InnerProductSpace.Basic
- Cited by
- 72 results in Mathlib
- Foundations
- Depth 168 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.
- Realproof · cited by 25,697
- Norm.normstatement and proof · cited by 5,413
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- Inner.innerstatement · cited by 1,089
- RCLike.ofRealstatement and proof · cited by 350
- RCLike.ofReal_powproof · cited by 9
- InnerProductSpace.norm_sq_eq_re_innerproof · cited by 7
- inner_self_ofReal_reproof · cited by 3
Cited by72
Results whose statement or proof uses this declaration.
- inner_self_eq_zeroproof · cited by 18
- real_inner_self_eq_norm_mul_normproof · cited by 17
- orthonormal_iff_iteproof · cited by 15
- Submodule.orthogonal_orthogonalproof · cited by 14
- InnerProductSpace.gramSchmidt_orthogonalproof · cited by 6
- innerSL_apply_normproof · cited by 5
- Dense.eq_zero_of_inner_leftproof · cited by 4
- Matrix.posSemidef_gramproof · cited by 4
- EuclideanGeometry.dist_smul_vadd_eq_distproof · cited by 4
- linearIndependent_of_ne_zero_of_inner_eq_zeroproof · cited by 3
- EuclideanGeometry.Sphere.secondInter_zeroproof · cited by 3
- norm_inner_eq_norm_tfaeproof · cited by 3