Theorems · Theorem · functional analysis

ContinuousLinearMap.smul_compLp

∀ {α : Type u_1} {E : Type u_4} {F : Type u_5} {m : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α}
  [inst : NormedAddCommGroup E] [inst_1 : NormedAddCommGroup F] {𝕜 : Type u_6} {𝕜' : Type u_7}
  [inst_2 : NontriviallyNormedField 𝕜] [inst_3 : NontriviallyNormedField 𝕜'] [inst_4 : NormedSpace 𝕜 E]
  [inst_5 : NormedSpace 𝕜' F] {σ : 𝕜 →+* 𝕜'} [inst_6 : RingHomIsometric σ] {𝕜'' : Type u_8} [inst_7 : NormedRing 𝕜'']
  [inst_8 : Module 𝕜'' F] [inst_9 : IsBoundedSMul 𝕜'' F] [inst_10 : SMulCommClass 𝕜' 𝕜'' F] (c : 𝕜'') (L : E →SL[σ] F)
  (f : ↥(MeasureTheory.Lp E p μ)), (c • L).compLp f = c • L.compLp f

Defined in

Mathlib.MeasureTheory.Function.LpSpace.Basic

Cited by

1 results in Mathlib

Foundations

Depth 232 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedAddCommGroupNontriviallyNormedFieldNontriviallyNormedFieldNormedSpaceNormedSpaceRingHomIsometricNormedRingModuleIsBoundedSMulSMulCommClass