instContinuousSMulDomMulAct — Mathlib

English

There is a continuous scalar action of Mᵈᵐᵃ on Lp(E,p,μ) in the DomMulAct setting.

Русский

Существуeет непрерывное скалярное действие Mᵈᵐᵃ на Lp(E,p,μ) в обстановке DomMulAct.

LaTeX

$$ContinuousSMul (DomMulAct M) (Lp E p μ)$$

Lean4

@[to_additive]
instance instContinuousSMulDomMulAct : ContinuousSMul Mᵈᵐᵃ (Lp E p μ) where
  continuous_smul :=
    let g : C(Mᵈᵐᵃ × Lp E p μ, C(X, X)) :=
      (ContinuousMap.mk (fun a : M × X ↦ a.1 • a.2) continuous_smul).curry.comp <|
        .comp (.mk DomMulAct.mk.symm) ContinuousMap.fst
    continuous_snd.compMeasurePreservingLp g.continuous _ Fact.out

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