/-- Riemann-Lebesgue lemma for functions on a finite-dimensional real vector space, formulated via
dual space. -/
theorem tendsto_integral_exp_smul_cocompact (μ : Measure V) [μ.IsAddHaarMeasure] :
    Tendsto (fun w : V →L[ℝ] ℝ => ∫ v, e[-w v] • f v ∂μ) (cocompact (V →L[ℝ] ℝ)) (𝓝 0) := by
  -- We have already proved the result for inner-product spaces, formulated in a way which doesn't
  -- refer to the inner product. So we choose an arbitrary inner-product space isomorphic to V
  -- and port the result over from there.
  let V' := EuclideanSpace ℝ (Fin (finrank ℝ V))
  have A : V ≃L[ℝ] V' := toEuclidean
  borelize V'
  -- various equivs derived from A
  let Aₘ : MeasurableEquiv V V' := A.toHomeomorph.toMeasurableEquiv
  -- isomorphism between duals derived from A -- need to do continuity as a separate step in order
  -- to apply `LinearMap.continuous_of_finiteDimensional`.
  let Adualₗ : (V →L[ℝ] ℝ) ≃ₗ[ℝ] V' →L[ℝ] ℝ :=
    { toFun := fun t => t.comp A.symm.toContinuousLinearMap
      invFun := fun t => t.comp A.toContinuousLinearMap
      map_add' := by
        intro t s
        ext1 v
        simp only [ContinuousLinearMap.coe_comp', Function.comp_apply,
          ContinuousLinearMap.add_apply]
      map_smul' := by
        intro x f
        ext1 v
        simp only [RingHom.id_apply, ContinuousLinearMap.coe_comp', Function.comp_apply,
          ContinuousLinearMap.smul_apply]
      left_inv := by
        intro w
        ext1 v
        

Proof state:
case h
E : Type u_1
V : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℂ E
f : V → E
inst✝¹⁰ : AddCommGroup V
inst✝⁹ : TopologicalSpace V
inst✝⁸ : TopologicalAddGroup V
inst✝⁷ : T2Space V
inst✝⁶ : MeasurableSpace V
inst✝⁵ : BorelSpace V
inst✝⁴ : Module ℝ V
inst✝³ : ContinuousSMul ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : CompleteSpace E
μ : Measure V
inst✝ : Measure.IsAddHaarMeasure μ
V' : Type := EuclideanSpace ℝ (Fin (finrank ℝ V))
A : V ≃L[ℝ] V'
this✝¹ : MeasurableSpace V' := borel V'
this✝ : BorelSpace V'
Aₘ : V ≃ᵐ V' := Homeomorph.toMeasurableEquiv (ContinuousLinearEquiv.toHomeomorph A)
w : V →L[ℝ] ℝ
v : V
⊢ ((fun t => ContinuousLinearMap.comp t ↑A)
        (AddHom.toFun
          {
              toAddHom :=
                { toFun := fun t => ContinuousLinearMap.comp t ↑(ContinuousLinearEquiv.symm A),
                  map_add' :=
                    (_ :
                      ∀ (t s : V →L[ℝ] ℝ),
                        (fun t => ContinuousLinearMap.comp t ↑(ContinuousLinearEquiv.symm A)) (t + s) =
                          (fun t => ContinuousLinearMap.comp t ↑(ContinuousLinearEquiv.symm A)) t +
                            (fun t => ContinuousLinearMap.comp t ↑(ContinuousLinearEquiv.symm A)) s) },
              map_smul' :=
                (_ :
                  ∀ (x : ℝ) (f : V →L[ℝ] ℝ),
                    AddHom.toFun
                        { toFun := fun t => ContinuousLinearMap.comp t ↑(ContinuousLinearEquiv.symm A),
                          map_add' :=
                            (_ :
                              ∀ (t s : V →L[ℝ] ℝ),
                                (fun t => ContinuousLinearMap.comp t ↑(ContinuousLinearEquiv.symm A)) (t + s) =
                                  (fun t => ContinuousLinearMap.comp t ↑(ContinuousLinearEquiv.symm A)) t +
                                    (fun t => ContinuousLinearMap.comp t ↑(ContinuousLinearEquiv.symm A)) s) }
                        (x • f) =
                      (RingHom.id ℝ) x •
                        AddHom.toFun
                          { toFun := fun t => ContinuousLinearMap.comp t ↑(ContinuousLinearEquiv.symm A),
                            map_add' :=
                              (_ :
                                ∀ (t s : V →L[ℝ] ℝ),
                                  (fun t => ContinuousLinearMap.comp t ↑(ContinuousLinearEquiv.symm A)) (t + s) =
                                    (fun t => ContinuousLinearMap.comp t ↑(ContinuousLinearEquiv.symm A)) t +
                                      (fun t => ContinuousLinearMap.comp t ↑(ContinuousLinearEquiv.symm A)) s) }
                          f) }.toAddHom
          w))
      v =
    w v

Next tactic:
simp only [ContinuousLinearMap.coe_comp',
          ContinuousLinearEquiv.coe_coe, Function.comp_apply,
          ContinuousLinearEquiv.symm_apply_apply]