Theorems · Theorem · functional analysis

inner_smul_left_eq_star_smul

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : SeminormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {𝕝 : Type u_4} [inst_3 : CommSemiring 𝕝] [inst_4 : StarRing 𝕝] [inst_5 : Algebra 𝕝 𝕜] [inst_6 : Module 𝕝 E]
  [IsScalarTower 𝕝 𝕜 E] [StarModule 𝕝 𝕜] (x y : E) (r : 𝕝), inner 𝕜 (r • x) y = (starRingEnd 𝕝) r • inner 𝕜 x y

See inner_smul_left for the common special when 𝕜 = 𝕝.

Defined in

Mathlib.Analysis.InnerProductSpace.Basic

Cited by

4 results in Mathlib

Foundations

Depth 47 from the axioms · uses propext, Quot.sound

Assumes

RCLikeSeminormedAddCommGroupInnerProductSpaceCommSemiringStarRingAlgebraModuleIsScalarTowerStarModule