Theorems · Theorem · functional analysis
inner_smul_right
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : SeminormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
(x y : E) (r : 𝕜), inner 𝕜 x (r • y) = r * inner 𝕜 x ySee inner_smul_right_eq_smul for the case of a general algebra action.
- Defined in
- Mathlib.Analysis.InnerProductSpace.Basic
- Cited by
- 68 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.
Cites5
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
- Inner.innerstatement · cited by 1,089
- inner_smul_right_eq_smulproof · cited by 3
Cited by69
Results whose statement or proof uses this declaration.
- innerₛₗproof · cited by 28
- real_inner_smul_rightproof · cited by 7
- InnerProductSpace.gramSchmidt_orthogonalproof · cited by 6
- InnerProductGeometry.angle_smul_right_of_posproof · cited by 6
- Dense.eq_zero_of_inner_leftproof · cited by 4
- Matrix.toLin_conjTransposeproof · cited by 4
- MeasureTheory.charFun_eq_fourierIntegral'proof · cited by 3
- Matrix.star_dotProduct_gram_mulVecproof · cited by 3
- Affine.Simplex.mongePoint_mem_mongePlaneproof · cited by 3
- LinearMap.IsSymmetric.orthogonalFamily_eigenspacesproof · cited by 3
- linearIndependent_of_ne_zero_of_inner_eq_zeroproof · cited by 3
- Orthonormal.inner_right_finsuppproof · cited by 3