Theorems · Theorem · functional analysis
inner_smul_left
∀ {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 (r • x) y = (starRingEnd 𝕜) r * inner 𝕜 x ySee inner_smul_left_eq_star_smul for the case of a general algebra action.
- Defined in
- Mathlib.Analysis.InnerProductSpace.Basic
- Cited by
- 49 results in Mathlib
- Foundations
- Depth 48 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- RingHomstatement · cited by 10,189
- 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
- starRingEndstatement · cited by 671
- inner_smul_left_eq_star_smulproof · cited by 4
Cited by50
Results whose statement or proof uses this declaration.
- inner_zero_leftproof · cited by 59
- inner_neg_leftproof · cited by 33
- innerₛₗproof · cited by 28
- real_inner_smul_leftproof · cited by 19
- Submodule.mem_orthogonal_singleton_iff_inner_rightproof · cited by 10
- Matrix.toLin_conjTransposeproof · cited by 4
- LinearMap.IsSymmetric.orthogonalFamily_eigenspacesproof · cited by 3
- Matrix.star_dotProduct_gram_mulVecproof · cited by 3
- inner_map_polarizationproof · cited by 2
- LinearMap.IsSymmetric.smulproof · cited by 2
- inner_smul_real_leftproof · cited by 2
- LinearMap.isPositive_natCastproof · cited by 2